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. :-)
