don't forgot signal processing. Complex number theory is used a lot when it comes down to Fourier transforms of discrete or continuous functions.

but yea the naming convention of "imaginary numbers" is pretty bad, we did the same thing for negative numbers (we called it imaginary) when people couldn't comprehend of a negative quantity.

negative numbers are defined from a conservative system, basically a gain/loss relationship: if I have 2 apples and I give away 1 to a friend, I have 1 apple left, that loss -- that apple I gave to a friend -- is the negative quantity. For imaginary numbers, we don't really have a way to comprehend an imaginary quantity; what does it mean that I have sqrt(-1) apples.