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.