Having mentioned this a few times in course reports, I thought I’d throw out a link to my lecture notes (PDF) on complex numbers. This is the one-class whirlwind review of complex numbers from defining i to Euler’s theorem about complex exponentials.
To answer a slightly incredulous question from a commenter, this is necessary because the math department does not teach about complex numbers exponentials (edited to correct an inadvertent slur against the math department) in the calculus sequence, and the only math prerequisites for the sophomore modern physics class I’m teaching are calculus classes (I don’t recall whether it’s Calc III or Calc IV, but that’s it). Most of our majors take more math than that, and so probably see complex numbers in a math context, but they don’t get it before my class, and I need to use complex exponentials when I talk about solutions of the Schrödinger Equation.
As for why the math department doesn’t teach this in the calculus sequence, I think it’s another curricular distortion caused by our trimester calendar. I’m not certain about that, though– you’d have to ask one of our mathematicians.