Demystifying complex numbers At the end of this month I start teaching complex analysis to
2nd year undergraduates, mostly from engineering but some from
science and maths. The main applications for them in future
studies are contour integrals and Laplace transform, but this should be a "real" complex analysis course which I could later
refer to in honours courses. I am now confident (after
this
discussion, especially Gauss’s complaints given in Keith’s comment)
that the name "complex" is discouraging to average students.
Why do we need to study numbers which do not belong to the real world?
We all know that the thesis is wrong and I have in mind some examples
where the use of complex variable functions simplify solving considerably
(I give two below). The drawback is that all them assume some
knowledge from students already.
So I would be happy to learn elementary examples which may
convince students that complex numbers and functions of a
complex variable are useful. As this question runs in the community wiki mode,
I would be glad to see one example per answer.
Thank you in advance!
Here are the two promised examples. I was reminded of the second one by several answers and comments about trigonometric functions (and also by the notification that "the bounty on your question Trigonometry related to Rogers–Ramanujan identities expires within three days"; it seems to be harder than I expected).
Example 1.
What is the Fourier expansion of the (unbounded) periodic function
$$
f(x)=\ln\Bigl\lvert\sin\frac x2\Bigr\rvert\ ?
$$
Solution.
The function $f(x)$ is periodic with period $2\pi$ and has poles at the
points $2\pi k$, $k\in\mathbb Z$.
Consider the function on the interval $x\in[\varepsilon,2\pi-\varepsilon]$.
The series
$$
\sum_{n=1}^\infty\frac{z^n}n, \qquad z=e^{ix},
$$
converges for all values $x$ from the interval.
Since
$$
\Bigl\lvert\sin\frac x2\Bigr\rvert=\sqrt{\frac{1-\cos x}2}
$$
and $\operatorname{Re}\ln w=\ln\lvert w\rvert$, where we choose $w=\frac12(1-z)$,
we deduce that
$$
\operatorname{Re}\Bigl(\ln\frac{1-z}2\Bigr)=\ln\sqrt{\frac{1-\cos x}2}
=\ln\Bigl\lvert\sin\frac x2\Bigr\rvert.
$$
Thus,
$$
\ln\Bigl\lvert\sin\frac x2\Bigr\rvert
=-\ln2-\operatorname{Re}\sum_{n=1}^\infty\frac{z^n}n
=-\ln2-\sum_{n=1}^\infty\frac{\cos nx}n.
$$
As $\varepsilon>0$ can be taken arbitrarily small,
the result remains valid for all $x\ne2\pi k$.
Example 2.
Let $p$ be an odd prime number.
$\newcommand\Legendre{\genfrac(){}{}}$For an integer $a$ relatively prime to $p$,
the Legendre symbol $\Legendre ap$ is $+1$ or $-1$
depending on whether the congruence
$x^2\equiv a\pmod{p}$ is solvable or not.
Using the elementary result (a consequence of Fermat's little theorem) that
$$
\Legendre ap \equiv a^{(p-1)/2}\pmod p,
\tag{*}\label{star}
$$
show that
$$
\Legendre 2p=(-1)^{(p^2-1)/8}.
$$
Solution.
In the ring $\mathbb Z+\mathbb Zi=\Bbb Z[i]$, the binomial formula implies
$$
(1+i)^p\equiv1+i^p\pmod p.
$$
On the other hand,
$$
(1+i)^p
=\bigl(\sqrt2e^{\pi i/4}\bigr)^p
=2^{p/2}\biggl(\cos\frac{\pi p}4+i\sin\frac{\pi p}4\biggr)
$$
and
$$
1+i^p
=1+(e^{\pi i/2})^p
=1+\cos\frac{\pi p}2+i\sin\frac{\pi p}2
=1+i\sin\frac{\pi p}2.
$$
Comparing the real parts implies that
$$
2^{p/2}\cos\frac{\pi p}4\equiv1\pmod p,
$$
hence from $\sqrt2\cos(\pi p/4)\in\{\pm1\}$ we conclude that
$$
2^{(p-1)/2}\equiv\sqrt2\cos\frac{\pi p}4\pmod p.
$$
Then using the elementary result \eqref{star}:
$$
\Legendre2p
\equiv2^{(p-1)/2}
\equiv\sqrt2\cos\frac{\pi p}4
=\begin{cases}
1 & \text{if } p\equiv\pm1\pmod8, \cr
-1 & \text{if } p\equiv\pm3\pmod8,
\end{cases}
$$
which is exactly the required formula.
 A: This is not exactly an answer to the question, but it is the simplest thing I know to help students appreciate complex numbers.  (I got the idea somewhere else, but I forgot exactly where.)
It's something even much younger students can appreciate.  Recall that on the real number line, multiplying a number by -1 "flips" it, that is, it rotates the point 180 degrees about the origin.  Introduce the imaginary number line (perpendicular to the real number line) then introduce multiplication by i as a rotation by 90 degrees.  I think most students would appreciate operations on complex numbers if they visualize them as movements of points on the complex plane.
A: From the perspective of complex analysis, the theory of Fourier series has a very natural explanation. I take it that the students had seen Fourier series first, of course. I had mentioned this elsewhere too. I hope the students also know about Taylor theorem and Taylor series. Then one could talk also of the Laurent series in concrete terms, and argue that the Fourier series is studied most naturally in this setting.
First, instead of cos and sin, define the Fourier series using complex exponential. Then, let $f(z)$ be a complex analytic function in the complex plane, with period $1$. 
Then write the substitution $q = e^{2\pi i z}$. This way the analytic function $f$ actually becomes a meromorphic function of $q$ around zero, and $z = i \infty$ corresponds to $q = 0$. The Fourier expansion of  $f(z)$ is then nothing but the Laurent expansion of $f(q)$ at $q = 0$.
Thus we have made use of a very natural function in complex analysis, the exponential function, to see the periodic function in another domain. And in that domain, the Fourier expansion is nothing but the Laurent expansion, which is a most natural thing to consider in complex analysis.
I am am electrical engineer; I have an idea what they all study; so I can safely override any objections that this won't be accessible to electrical engineers. Moreover, the above will reduce their surprise later in their studies when they study signal processing and wavelet analysis.
A: Another classic I haven't seen mentioned yet is the proof of the Machin formula
$$
\frac{\pi}{4} = 4\arctan \frac{1}{5}-\arctan \frac{1}{239}.
$$
I honestly don't know a proof of this that avoids complex numbers, but surely it can be nowhere near as simple as the elementary computation needed to prove the identity
$$
2+2i = \frac{(5+i)^4}{239+i}.
$$
Taking $\arg$ on both sides yields Machin's formula.
A: You can solve the differential equation y''+y=0 using complex numbers. Just write 
$$(\partial^2 + 1) y = (\partial +i)(\partial -i) y$$
and you are now dealing with two order one differential equations
that are easily solved 
$$(\partial +i) z =0,\qquad (\partial -i)y =z$$
The multivariate case is a bit harder and uses quaternions or Clifford algebras. This was done by Dirac for the Schrodinger equation ($-\Delta \psi = i\partial_t \psi$), and that led him to the prediction of the existence of antiparticles (and to the Nobel prize).
A: Maybe artificial, but a nice example (I think) demonstrating analytic continuation (NOT just the usual $\mathrm{Re}(e^{i \theta})$ method!) I don't know any reasonable way of doing this by real methods.
As a fun exercise, calculate 
$$
I(\omega) = \int_0^\infty e^{-x} \cos (\omega x) \frac{dx}{\sqrt{x}}, \qquad \omega \in \mathbb{R}
$$
from the real part of $F(1+i \omega)$, where 
$$
F(k) = \int_0^\infty e^{-kx} \frac{dx}{\sqrt{x}}, \qquad \mathrm{Re}(k)>0
$$
(which is easily obtained for $k>0$ by a real substitution) and using analytic continuation to justify the same formula with $k=1+i \omega$.
You need care with square roots, branch cuts, etc.; but this can be avoided by considering $F(k)^2$, $I(\omega)^2$.
Of course all the standard integrals provide endless fun examples! (But the books don't have many requiring genuine analytic continuation like this!)
A: Consider the function f(x)=1/(1+x^2) on the real line. Using the geometric progression
formula, you can expand f(x)=1-x^2+... .
This series converges for |x|<1 but diverges for all other x.
Why this is so? The function looks nice and smooth everywhere on the real line.
This example is taken from the Introduction of the textbook by B. V. Shabat.
A: This answer doesn't show how the complex numbers are useful, but I think it might demystify them for students. Most are probably already familiar with its content, but it might be useful to state it again. Since the question was asked two months ago and Professor Zudilin started teaching a month ago, it's likely this answer is also too late.
If they have already taken a class in abstract algebra, one can remind them of the basic theory of field extensions with emphasis on the example of $\mathbb C \cong \mathbb R[x]/(x^2+1).$
It seems that most introductions give complex numbers as a way of writing non-real roots of polynomials and go on to show that if multiplication and addition are defined a certain way, then we can work with them, that this is consistent with handling them like vectors in the plane, and that they are extremely useful in solving problems in various settings. This certainly clarifies how to use them and demonstrates how useful they are, but it still doesn't demystify them. A complex number still seems like a magical, ad hoc construction that we accept because it works. If I remember correctly, and has probably already been discussed, this is why they were called imaginary numbers.
If introduced after one has some experience with abstract algebra as a field extension, one can see clearly that the complex numbers are not a contrivance that might eventually lead to trouble. Beginning students might be thinking this and consequently, resist them, or require them to have faith in them or their teachers, which might already be the case. Rather, one can see that they are the result of a natural operation. That is, taking the quotient of a polynomial ring over a field and an ideal generated by an irreducible polynomial, whose roots we are searching for.
Multiplication, addition, and its 2-dimensional vector space structure over the reals are then consequences of the quotient construction $\mathbb R[x]/(x^2+1).$ The root $\theta,$ which we can then relabel to $i,$ is also automatically consistent with familiar operations with polynomials, which are not ad hoc or magical. The students should also be able to see that the field extension $\mathbb C = \mathbb R(i)$ is only one example, although a special and important one, of many possible quotients of polynomial rings and maximal ideals, which should dispel ideas of absolute uniqueness and put it in an accessible context. Finally, if they think that complex numbers are imaginary, that should be corrected when they understand that they are one example of things naturally constructed from other things they are already familiar with and accept.
Reference: Dummit & Foote: Abstract Algebra, 13.1
A: Students usually find the connection of trigonometric identities like $\sin(a+b)=\sin a\cos b+\cos a\sin b$    to multiplication of complex numbers striking.
A: Try this: compare the problems of finding the points equidistant in the plane from (-1, 0) and (1, 0),  which is easy, with finding the points at twice the distance from (-1, 0) that they are from (1, 0). The idea that "real" concepts are the only ones of use in the "real world" is of course a fallacy. I suppose it is more than a century since electrical engineers admitted that complex numbers are useful.
A: I always like to use complex dynamics to illustrate that complex numbers are "real" (i.e., they are not just a useful abstract concept, but in fact something that very much exist, and closing our eyes to them would leave us not only devoid of useful tools, but also of a deeper understanding of phenomena involving real numbers.) Of course I am a complex dynamicist so I am particularly partial to this approach!
Start with the study of the logistic map $x\mapsto \lambda x(1-x)$ as a dynamical system (easy to motivate e.g. as a simple model of population dynamics). Do some experiments that illustrate some of the behaviour in this family (using e.g. web diagrams and the Feigenbaum diagram), such as:


*

*The period-doubling bifurcation

*The appearance of periodic points of various periods

*The occurrence of "period windows" everywhere in the Feigenbaum diagram.


Then let x and lambda be complex, and investigate the structure both in the dynamical and parameter plane, observing


*

*The occurence of beautiful and very "natural"-looking objects in the form of Julia sets and the (double) Mandelbrot set;

*The explanation of period-doubling as the collision of a real fixed point with a complex point of period 2, and the transition points occuring as points of tangency between interior components of the Mandelbrot set;

*Period windows corresponding to little copies of the Mandelbrot set.


Finally, mention that density of period windows in the Feigenbaum diagram - a purely real result, established only in the mid-1990s - could never have been achieved without complex methods.
There are two downsides to this approach:
* It requires a certain investment of time; even if done on a superficial level (as I sometimes do in popular maths lectures for an interested general audience) it requires the better part of a lecture
* It is likely to appeal more  to those that are mathematically minded than engineers who could be more impressed by useful tools for calculations such as those mentioned elsewhere on this thread.
However, I personally think there are few demonstrations of the "reality" of the complex numbers that are more striking. In fact, I have sometimes toyed with the idea of writing an introductory text on complex numbers which uses this as a primary motivation.
A: "Why do we need to study numbers which do not belong to the real world?"
I don't think you can answer this in a single class.  The best answer I can come up with is to show how complicated calculus problems can be solved easily using complex analysis.
As an example, I bet most of your students hated solving the problem $\int e^{-x}\cos(x) dx$.  Solve it for them the way they learned it in calculus, by repeated integration by parts and then by $\int e^{-x}\cos(x) dx\ \ =\ \ \Re \int e^{-x(1-i)}dx$.  They should notice how much easier it was to use complex analysis. If you do this enough they might come to appreciate numbers that do not belong to the real world.
A: Motivating complex analysis
The physics aspect of motivation should be the strongest for engineering students.  No complex numbers, no quantum mechanics, no solid state physics, no lasers, no electrical or electronic engineering (starting with impedance), no radio, TV, acoustics, no good simple way of understanding of the mechanical analogues of RLC circuits, resonance, etc., etc.
Then the "mystery" of it all.  Complex numbers as the consequence  of roots, square, cubic, etc., unfolding until one gets the complex plane, radii of convergence, poles of stability, all everyday engineering.    Then the romance of it all, the "self secret knowledge", discovered over hundreds of years, a new language which even helps our thinking in general.   Then the wider view of say Smale/Hirsch on higher dimensional differential equations, chaos etc.    They should see the point pretty quickly.  This is a narrow door, almost accidentally discovered, through which we see and understand entire new realms, which have become our best current, albeit imperfect,descriptions of  how to understand and manipulate a kind of "inner essence of what is" for practical human ends, i.e. engineering.   (True, a little over the top, but then pedagogical and motivational).   
For them to say that they just want to learn a few computational tricks is a little like a student saying, "don't teach me about fire, just about lighting matches".   It's up to them I suppose, but they will always be limited.
There might be some computer software engineer who needs a little more, but then I suppose there is also modern combinatorics.  :-)
A: "Why do we need to study numbers which do not belong to the real world?"
Having been through the relevant mathematical mill, I subsequently engaged with Geometric Algebra (a Clifford Algebra interpreted strictly geometrically).
Once I understood that the square of a unit bivector is -1 and then how rotors worked, all my (conceptual) difficulties evaporated.
I have never had a reason to use (pure) complex numbers since and I suspect that most engineering/physics/computing types would avoid them if they were able.
Likely you have the above group mixed together with pure mathematicians that feel quite at home with the non-physical aspects of complex numbers and wouldn't dream of asking such an impertinent question:-)
A: From "Birds and Frogs" by Freeman Dyson [Notices of Amer. Math. Soc. 56 (2009) 212--223]:

One of the most profound jokes of
  nature is the square root of minus one
  that the physicist Erwin
  Schrödinger put into his wave
  equation when he invented wave
  mechanics in 1926. Schrödinger
  was a bird who started from the idea
  of unifying mechanics with optics. A
  hundred years earlier, Hamilton had
  unified classical mechanics with ray
  optics, using the same mathematics to
  describe optical rays and classical
  particle trajectories.
  Schrödinger’s idea was to extend
  this unification to wave optics and
  wave mechanics. Wave optics already
  existed, but wave mechanics did not.
  Schrödinger had to invent wave
  mechanics to complete the unification.
  Starting from wave optics as a model,
  he wrote down a differential equation
  for a mechanical particle, but the
  equation made no sense. The equation
  looked like the equation of conduction
  of heat in a continuous medium. Heat
  conduction has no visible relevance to
  particle mechanics. Schrödinger’s
  idea seemed to be going nowhere. But
  then came the surprise.
  Schrödinger put the square root
  of minus one into the equation, and
  suddenly it made sense. Suddenly it
  became a wave equation instead of a
  heat conduction equation. And
  Schrödinger found to his delight
  that the equation has solutions
  corresponding to the quantized orbits
  in the Bohr model of the atom. It
  turns out that the Schrödinger
  equation describes correctly
  everything we know about the behavior
  of atoms. It is the basis of all of
  chemistry and most of physics. And
  that square root of minus one means
  that nature works with complex numbers
  and not with real numbers. This
  discovery came as a complete surprise,
  to Schrödinger as well as to
  everybody else. According to
  Schrödinger, his
  fourteen-year-old girl friend Itha
  Junger said to him at the time, "Hey,
  you never even thought when you began
  that so much sensible stuff would come
  out of it." All through the nineteenth
  century, mathematicians from Abel to
  Riemann and Weierstrass had been
  creating a magnificent theory of
  functions of complex variables. They
  had discovered that the theory of
  functions became far deeper and more
  powerful when it was extended from
  real to complex numbers. But they
  always thought of complex numbers as
  an artificial construction, invented
  by human mathematicians as a useful
  and elegant abstraction from real
  life. It never entered their heads
  that this artificial number system
  that they had invented was in fact the
  ground on which atoms move. They never
  imagined that nature had got there
  first.

A: An interesting example of usage of complex numbers can be found in
http://arxiv.org/abs/math/0001097 (Michael Eastwood, Roger Penrose, Drawing with Complex Numbers).
A: From the point of view of enginieers, the most obvious application of complex numbers is computing alternating currents.
Consider first direct current. If you have a network of resistors, and want to compute the current in this network, or the potential of a node, then Kirchhoff's rules reduce this problem to a system of linear equations. Kirchhoff's rules are obvious, essentially saying that ellectric current cannot just disappear.
If you have alternating current, you have capacities and inductions in addition to the resistors, but if you consider them as imaginary resistance depending on the frequency, the computations are exactly the same as in the direct case, just over another field. The alternative would be computing the phase shift separately from the current, which is much more effort and only works for very simple networks, e.g. oscillators.
Once you learned Fourier analysis, this approach immediately tells you how a filter works, and whether a given network acts as a filter.
A: If the students have had a first course in differential equations, tell them to solve the system
$$x'(t) = -y(t)$$
$$y'(t) = x(t).$$
This is the equation of motion for a particle whose velocity vector is always perpendicular to its displacement.  Explain why this is the same thing as
$$(x(t) + iy(t))' = i(x(t) + iy(t))$$
hence that, with the right initial conditions, the solution is
$$x(t) + iy(t) = e^{it}.$$
On the other hand, a particle whose velocity vector is always perpendicular to its displacement travels in a circle.  Hence, again with the right initial conditions, $x(t) = \cos t, y(t) = \sin t$.  (At this point you might reiterate that complex numbers are real $2 \times 2$ matrices, assuming they have seen this method for solving systems of differential equations.)
A: Is it too abstract to motivate complex numbers in terms of the equations we can solve depending on whether we choose to work in ${\mathbb N, \mathbb Z, \mathbb Q, \mathbb R, \mathbb C}$?
The famous "John and Betty" (Link) takes such an approach.
A: As an example to demonstrate the usefulness of complex analysis in mechanics (which may seem
counterintuitive to engineering students, since mechanics is introduced on the reals), 
one may consider the simple problem of the one dimensional harmonic oscillator, whose Hamiltonian equations of motion are diagonalized in the complex representation, equivalently one needs to integrate a single (holomorphic) first order ODE instead of 
a single second order or two first order ODEs.
A: Here's a visual thing I handed out to students in a much more elementary class than the one the question mentions:
http://web.archive.org/web/20130701140646/http://www.math.umn.edu/~hardy/1031/handouts/March.3.pdf
A: It seems that many of the answers fall into two categories: those that are applications of Euler's formula $e^{i\theta}=\cos\theta + i\sin\theta$, and those that are related to the fundamental theorem of algebra.
To supply one other instance of the latter, I'd go with Bézout's theorem from algebraic geometry (allegedly one of Gauss's earlier proofs of the FTA was by way of Bézout's, which smells like circular reasoning).
The students may remember conics, or at least circles. How many points can a circle intersect with a line? 0, 1, or 2. What about an ellipse with a line? Same. Two ellipses? Up to 4. How come two circles never intersect in 3 or 4 points. Well it may require a little bit of homogeneous coordinates, but we may cheat by saying that the points $(\infty, i\infty)$ and $(\infty, -i\infty)$ are on every circle. It may help to compare with the situation of two hyperbolas whose asymptotes have slopes $\pm 1$.
A: Since the audience for such a class must be to a large degree E.E.s and M.E.s, I think they would appreciate a Reader's Digest version of the paper "Steinmetz and the Concept of Phasor: A Forgotten Story" by A. E. A. Araújo & D. A. V. Tonidandel.
For the human interest aspect of the story, there is much material from which to choose some vignettes. Biographies (videos (see comment), papers, books) of the two main characters in the paper—Steinmetz and Heaviside—contain many very entertaining technically-related tales of the two eccentrics, and some background on their contributions to the development of signal transmission via cables and the ether in the case of Heaviside and of the electric motor in the case of Steinmetz would accentuate the importance of their ideas, along with an overall portrait of Electric City, at which Steinmetz was the acclaimed wizard, and the laying of the trans-Atlantic cables, in which Heaviside played an important role, despite obstructions created by some authorities.
This epoch of engineering—the genesis of the age of electricity—lies between that of the steam engine and of the computer and IT. I think the engineers would appreciate both the technical and historical perspectives. (You could also slip in the importance of securing patents for financial security—every engineer has an eye on that—using H & S as contrasting examples.)
A: Here are two simple uses of complex numbers that I use to
try to convince students that complex numbers are "cool" and 
worth learning. 


*

*(Number Theory) Use complex numbers to derive 
Brahmagupta's identity expressing $(a^2+b^2)(c^2+d^2)$
as the sum of two squares, for integers $a,b,c,d$. 

*(Euclidean geometry) Use complex numbers to explain Ptolemy's
Theorem. For a cyclic quadrilateral with vertices $A,B,C,D$ we have 
$$\overline{AC}\cdot \overline{BD}=\overline{AB}\cdot \overline{CD}
+\overline{BC}\cdot \overline{AD}$$
A: One cannot over-emphasize that passing to complex numbers often permits a great simplification by linearizing what would otherwise be more complex nonlinear phenomena. One example familiar to any calculus student is the fact that integration of rational functions is much simpler over $\mathbb C$ (vs. $\mathbb R$) since partial fraction decompositions involve at most linear (vs quadratic) polynomials in the denominator. Similarly one reduces higher-order constant coefficient differential and difference equations to linear (first-order) equations by factoring the linear operators over $\mathbb C$. More generally one might argue that such simplification by linearization was at the heart of the development of abstract algebra. Namely, Dedekind, by abstracting out the essential linear structures (ideals and modules) in number theory, greatly simplified the prior nonlinear theory based on quadratic forms. This enabled him to exploit to the hilt the power of linear algebra. Examples abound of the revolutionary power that this brought to number theory and algebra - e.g. for one little-known gem see my recent post explaining how Dedekind's notion of conductor ideal beautifully encapsulates the essence of elementary irrationality proofs of n'th roots.
A: One of my favourite elementary applications of complex analysis is the evaluation of infinite sums of the form
$$\sum_{n\geq 0} \frac{p(n)}{q(n)}$$
where $p,q$ are polynomials and $\deg q > 1 + \deg p$, by using residues.
A: Complex numbers make working with polynomials much easier, including purely real-valued questions about polynomials.
For example, prove every real polynomial factors into linear and quadratics. With Fundamental Theorem of Algebra this is trivial (factors into complex linear polynomials; each complex root appears with its conjugate). A purely real proof would be ugly.
In my opinion, negative numbers do not "exist" either. However, negative numbers make subtraction much easier. Complex numbers fill the analogous role for algebraic manipulations.
A: Start with a line segment AB, and pick a point P on this.
On AP and PB, create two equilateral triangles. You get two new points, A2 and B2. Put P2 on the line between these, such that the
ratios $AP:PB$ and $A_2P_2:P_2B_2$ are the same. Repeat.
This has a limit point, $P^*$. Show that this $P^*$ lie on a circle with AB as diameter.
This is a neat problem, where complex numbers do help quite a lot; although one can of course use linear algebra also.

A: I noticed this old question because it got bumped recently, and am surprised that the original, historical motivation for complex numbers—namely, a formula for solving a cubic equation—does not seem to be mentioned in any of the answers.  There is now a lovely Veritasium video that tells the story in a way that I think would appeal to any second-year undergraduate student.
Complex numbers are not well-motivated by the quadratic equation, because their appearance just indicates the absence of (real) solutions, so you can dismiss them.  But when you try to find a formula for the cubic equation in terms of radicals, complex numbers unavoidably force themselves upon you even when the solutions are real.
A: If you really want to "demystify" complex numbers, I'd suggest teaching what complex multiplication looks like with the following picture, as opposed to a matrix representation:  
If you want to visualize the product "z w", start with '0' and 'w' in the complex plane, then make a new complex plane where '0' sits above '0' and '1' sits above 'w'.  If you look for 'z' up above, you see that 'z' sits above something you name 'z w'.  You could teach this picture for just the real numbers or integers first -- the idea of using the rest of the points of the plane to do the same thing is a natural extension.
You can use this picture to visually "demystify" a lot of things:


*

*Why is a negative times a negative a positive?  --- I know some people who lost hope in understanding math as soon as they were told this fact

*i^2 = -1  

*(zw)t = z(wt) --- I think this is a better explanation than a matrix representation as to why the product is associative

*|zw| = |z| |w| 

*(z + w)v = zv + wv

*The Pythagorean Theorem: draw (1-it)(1+it) = 1 + t^2
etc.


One thing that's not so easy to see this way is the commutativity (for good reasons).
After everyone has a grasp on how complex multiplication looks, you can get into the differential equation: $\frac{dz}{dt} = i z , z(0) = 1$
which Qiaochu noted travels counterclockwise in a unit circle at unit speed.  You can use it to give a good definition for sine and cosine -- in particular, you get to define $\pi$ as the smallest positive solution to $e^{i \pi} = -1$.  It's then physically obvious (as long as you understand the multiplication) that $e^{i(x+y)} = e^{ix} e^{iy}$, and your students get to actually understand all those hard/impossible to remember facts about trig functions (like angle addition and derivatives) that they were forced to memorize earlier in their lives.  It may also be fun to discuss how the picture for $(1 + \frac{z}{n})^n$ turns into a picture of that differential equation in the "compound interest" limit as $n \to \infty$; doing so provides a bridge to power series, and gives an opportunity to understand the basic properties of the real exponential function more intuitively as well.
But this stuff is less demystifying complex numbers and more... demystifying other stuff using complex numbers.
Here's a link to some Feynman lectures on Quantum Electrodynamics (somehow prepared for a general audience) if you really need some flat out real-world complex numbers
http://video.google.com.au/videosearch?q=feynman+auckland&filter=0&start=0#
A: They're useful just for doing ordinary geometry when programming.
A common pattern I have seen in a great many computer programs is to start with a bunch of numbers that are really ratios of distances. Theses numbers get converted to angles with inverse trig functions. Then some simple functions are applied to the angles and the trig functions are used on the results.
Trig and inverse trig functions are expensive to compute on a computer. In high performance code you want to eliminate them if possible. Quite often, for the above case, you can eliminate the trig functions. For example $\cos(2\cos^{-1} x) = 2x^2-1$ (for $x$ in a suitable range) but the version on the right runs much faster.
The catch is remembering all those trig formulae. It'd be nice to make the compiler do all the work. A solution is to use complex numbers. Instead of storing $\theta$ we store $(\cos\theta,\sin\theta)$. We can add angles by using complex multiplication, multiply angles by integers and rational numbers using powers and roots and so on. As long as you don't actually need the numerical value of the angle in radians you need never use trig functions. Obviously there comes a point where the work of doing operations on complex numbers may outweigh the saving of avoiding trig. But often in real code the complex number route is faster.
(Of course it's analogous to using quaternions for rotations in 3D. I guess it's somewhat in the spirit of rational trigonometry except I think it's easier to work with complex numbers.)
A: Several motivating physical applications are listed on wikipedia 

Why do we need to study numbers which
  do not belong to the real world?

You may want to stoke the students' imagination by disseminating the deeper truth - that the world is neither real, complex nor p-adic (these are just completions of Q).  Here is a nice quote by Yuri Manin picked from here
On the fundamental level our world is neither real nor p-adic; it is adelic. For some reasons, reflecting the physical nature of our kind of living matter (e.g. the fact that we are built of massive particles), we tend to project the adelic picture onto its real side. We can equally well spiritually project it upon its non-Archimediean side and calculate most important things arithmetically.
The relations between "real" and "arithmetical" pictures of the world is that of complementarity, like the relation between conjugate observables in quantum mechanics.
(Y. Manin, in Conformal Invariance and String Theory, (Academic Press, 1989) 293-303 )
A: This is a specific example where complex numbers aid a task in elementary real analysis; I haven't thought about the extent to which it generalizes.
In my first year, I was given the task of formally proving that the Taylor series for arctan is
$$
\arctan(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \ldots,
$$ 
where "formally" meant not simply integrating the series for $1/(1+x^2)$ termwise, since we hadn't yet seen any theorems that said you could do that. We had, however, seen Taylor's theorem.
Hence the problem was to determine the values of all derivatives of $f(x)=\arctan(x)$, or of $f'(x)=1/(1+x^2)$, at $x=0$. However, it's not so easy to find a closed-form expression for the $n$-th derivative of $1/(1+x^2)$, unless you write it as
$$
f'(x) = \frac{1}{2i} \left( \frac{1}{x-i}-\frac{1}{x+i} \right),
$$ 
which then immediately yields
$$
f^{(n)}(x) = \frac{(-1)^{n-1}  (n-1)!}{2i} \cdot \left( \frac{1}{(x-i)^n}-\frac{1}{(x+i)^n} \right)
$$
for $n>0$, which then gives the answers $f^{(2n)}(0)=0$ and $f^{(2n+1)}(0)=(-1)^n \cdot (2n)!$. Combining this with Taylor's theorem gives the desired series.
I still think this is pretty neat. There really isn't any obvious way to cut the complex numbers and still have as painless a calculation as the one above.
A: *

*If they have a suitable background in linear algebra, I would not omit the interpretation of complex numbers in terms of conformal matrices of order 2 (with nonnegative determinant),   translating all operations on complex numbers (sum, product, conjugate, modulus, inverse) in the context of matrices: with special emphasis on their multiplicative action on the plane (in particular, "real" gives "homotety" and "modulus 1" gives "rotation").

*The complex exponential, defined initially as limit of $(1+z/n)^n$, should be a good application of the above geometrical ideas. In particular, for $z=it$, one can give a nice interpretation of the (too often covered with mystery) equation $e^{i\pi}=-1$ in terms of the length of the curve $e^{it}$ (defined as classical total variation).

*A brief discussion on (scalar) linear ordinary differential equations of order 2, with constant coefficients, also provides a good motivation (and with some historical truth).  

*Related to the preceding point, and especially because they are from engineering, it should be worth recalling all the useful complex formalism used in Electricity.   

*Not on the side of "real world" interpretation, but rather on the side of "useful abstraction" a brief account of the history of the third degree algebraic equation, with the embarrassing "casus impossibilis" (three real solutions, and the solution formula gives none, if taken in terms of "real" radicals!) should be very instructive. Here is also the source of such terms as "imaginary".
A: In answer to
"Why do we need to study numbers which do not belong to the real world?"
you might simply state that quantum mechanics tells us that complex numbers arise naturally in the correct description of probability theory as it occurs in our (quantum) universe.  
I think a good explanation of this is in Chapter 3 of the third volume of the Feynman lectures of physics, although I don't have a copy handy to check.  (In particular, similar to probability theory with real numbers, the complex amplitude of one of two independent events A or B occuring is just the sum of the amplitude of A and the amplitude of B. Furthermore, the complex amplitude of A followed by B is just the product of the amplitudes.  After all intermediate calculations one just takes the magnitude of the complex number squared to get the usual (real number) probability.)
A: This answer is an expansion of the answer of Yuri Bakhtin. 
Here is a kind of mime show.
Silently write the formulas for $\cos(2x)$ and $\sin(2x)$ lined up on the board, something like this:
$$\cos(2x) = \cos^2(x) \hphantom{+ 2 \cos(x) \sin(x)} - \sin^2(x)
$$
$$\sin(2x) = \hphantom{\cos^2(x)} + 2 \cos(x) \sin(x) \hphantom{- \sin^2(x)}
$$
Do the same for the formulas for $\cos(3x)$ and $\sin(3x)$, and however far you want to go:
$$\cos(3x) = \cos^3(x) \hphantom{+ 3 \cos^2(x) \sin(x)} - 3 \cos(x) \sin^2(x) \hphantom{- \sin^3(x)}
$$
$$\sin(3x) = \hphantom{\cos^3(x)} + 3 \cos^2(x) \sin(x) \hphantom{- 3 \cos(x) \sin^2(x)} - \sin^3(x)
$$
Maybe then let out a loud noise like "hmmmmmmmmm... I recognize those numbers..."
Then, on a parallel board, write out Pascal's triangle, and parallel to that write the application of Pascal's triangle to the binomial expansions $(x+y)^n$. Make some more puzzling sounds regarding those pesky plus and minus signs.
Then maybe it's time to actually say something: "Eureka! We can tie this all together by use of an imaginary number $i = \sqrt{-1}$". Then write out the binomial expansion of
$$(\cos(x) + i\,\sin(x))^n
$$
break it into its real and imaginary parts, and demonstrate equality with
$$\cos(nx) + i\, \sin(nx).
$$
A: The nicest elementary illustration I know of the relevance of complex numbers to calculus 
is its link to radius of convergence, which student learn how to compute by various tests, but more mechanically than conceptually.  The series for $1/(1-x)$, $\log(1+x)$, and $\sqrt{1+x}$ have radius of convergence 1 and we can see why: there's a problem at one of the endpoints of the interval of convergence (the function blows up or it's not differentiable).  However, 
the function $1/(1+x^2)$ is nice and smooth on the whole real line with no apparent problems, but its radius of convergence at the origin is 1.  From the viewpoint of real analysis this is strange: why does the series stop converging?  Well, if you look at distance 1 in the complex plane... 
More generally, you can tell them that for any rational function $p(x)/q(x)$, in reduced form, the radius of convergence of this function at a number $a$ (on the real line) is precisely the distance from $a$ to the nearest zero of the denominator, even if that nearest zero is not real. In other words, to really understand the radius of convergence in a general sense you have to work over the complex numbers.  (Yes, there are subtle distinctions between smoothness and analyticity which are relevant here, but you don't have to discuss that to get across the idea.) 
Similarly, the function $x/(e^x-1)$ is smooth but has a finite radius of convergence $2\pi$ (not sure if you can make this numerically apparent).  Again, on the real line the reason for this is not visible, but in the complex plane there is a good explanation.
A: I never took a precalculus class because every identity I've ever needed involving sines and cosines I could derive by evaluating a complex exponential in two different ways. Perhaps you could tell them that if they ever forget a trig identity, they can rederive it using this method?
A: Tristan Needham's book Visual Complex Analysis is full of these sorts of gems.  One of my favorites is the proof using complex numbers that if you put squares on the sides of a quadralateral, the lines connecting opposite centers will be perpendicular and of the same length.  After proving this with complex numbers, he outlines a proof without them that is much longer.
The relevant pages are on Google books: http://books.google.com/books?id=ogz5FjmiqlQC&lpg=PP1&dq=visual%20complex%20analysis&pg=PA16#v=onepage&q&f=false
A: How about how the holomorphicity of a function $f(z)=x+yi$ relates to, e.g., the curl of the vector $(x,y)\in\mathbb{R}^2$?  This relates nicely to why we can solve problems in two dimensional electromagnetism (or 3d with the right symmetries) very nicely using "conformal methods."  It would be very easy to start a course with something like this to motivate complex analytic methods.
A: This is a bit more about justifying the use of complex analytic functions, rather than complex numbers themselves, and so is perhaps jumping ahead too far for an initial introduction. But I would simply point out a few of the properties of differentiation versus integration of real functions.
On the one hand, derivatives are easy to compute, because they only depend on the local behavior of a function. The product rule and chain rule give us everything we need to compute derivatives of basically anything, provided we can write it in terms of familiar functions.
On the other hand, it is very difficult to make estimates or approximations of $f'$ based purely on estimates of $f$. By itself, an estimate of the form $f(x) \le g(x)$ allows us to say absolutely nothing about $f'$.
With integrals we have the opposite problem. A bound of the form $f(x) \le g(x)$ implies that $\int\limits_{0}^x f(t) dt \le \int\limits_{0}^x g(t) dt$. This enables us to approximate integrals of even complicated functions using simple bounds. However, the global nature of integration makes computing exact integrals or antiderivatives much harder, and there is no general procedure to find closed form antiderivatives of unfamiliar functions other than by inverting the product or chain rule, which only works in special cases.
In complex analysis, as opposed to real analysis, differentiation of $f$ is expressed as an integral of $f$. This means that bounds on $f$ do indeed allow us to bound the derivatives of $f$, giving us refined information about $f$'s local behavior. Conversely, we can evaluate integrals, which inherently depend on the global behavior of $f$ on the domain of integration, in terms of $f$'s local behavior. We can compute even complicated integrals just by evaluating some explicit function (or residue) at a point. Complex analysis lacks the disadvantages of the pure locality of real differentiation, and the pure globality of real integration, so both derivatives and integrals are much richer.
For emphasis you can give them an example of an integral which would be very difficult to compute with real methods, and tell them that soon they will be able to calculate it exactly.
A: I'm surprised nobody mentioned the way complex numbers actually "came to existence", namely for finding roots of third order polynomials, even when said roots are real.
A: Complex-step methods can be used to numerically estimate the derivative of real analytic functions with more numerical stability than real-step methods. Consider an analytic function $f(x+iy) = u + iv$.
By Cauchy-Riemann equations,
$$
\begin{align}
\frac{\partial u}{\partial x} & = \frac{\partial v}{\partial y} \\
& = \lim_{h \rightarrow 0} \frac{v(x+i(y+h))-v(x+iy)}{h}
\end{align}
$$
Since we are interested in the real parts of the function, $y = 0$, $u(x) = f(x)$ and $v(x) = 0$. Thus,
$$
\begin{align}
\frac{\partial f}{\partial x} &= \lim_{h \rightarrow 0} \frac{Im[f(x+ih)]}{h} \\
& \approx \frac{Im[f(x+ih)]}{h} \text{ (for small h) }
\end{align}
$$
The advantage to the real counterparts e.g.
$$
\begin{align}
\frac{df}{dx} &= \lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \\
& \approx \frac{f(x+h)-f(x)}{h} \text{ (for small h) }
\end{align}
$$
is that the subtraction in the numerator is eliminated. Thus, the complex-step method avoid catastrophic numerical cancellation (which usually happens when $h$ decreases too much and $f(x+h) - f(x)$ goes below the machine precision).
More details about the complex-step method can be seen in Martins, Joaquim R. R. A.; Sturdza, Peter; Alonso, Juan J., The complex-step derivative approximation, ACM Trans. Math. Softw. 29, No. 3, 245-262 (2003). ZBL1072.65027.
A: This video is a good teaser for an introductory course for students majoring in E.E., M.E., C.S., or physics:
What is a fourier series? From heat flow to drawing circles.
A: I think the description of complex numbers via the Argand Diagram is great. And we can see addition of complex numbers as translation and multiplication as a combination of rotation and dilation.
