I'm teaching a college-sophomore-level class in Oscillations and Waves and
started the semester by introducing that miracle formula e^i*theta = cos(theta)
+ i*sin(theta) for its usefulness in solving differential equations that
oscillate.
Probably not what you were looking for... But it reminds me of something you
could help with. "Real" and "Imaginary" are taken too literally by most
students. I have a tough time convincing them that the imaginary part of the
complex exponential is just as good a solution as the real part. "Real" and
"Imaginary" are just historical terms and don't mean "physical" and
"unphysical". I suppose this is the price we pay for rejecting complex roots of
polynomials by saying "Well they are Imaginary!, that's why."