I see. While it is probably true that the number of people who use it for AC circuit analysis and other applications for which the rotation analogy is appropria

I see. While it is probably true that the number of people who use it for AC circuit analysis and other applications for which the rotation analogy is appropriate is rather large, complex numbers are not "fundamentally about" rotation. In fact, the reason they're called "imaginary numbers" makes perfect sense if you were a mathematician 500 years ago - one method of solving cubic equation manipulated the square root of negative numbers in intermediate steps, which of course doesn't make sense, but by the final step these square roots canceled out and the final answer was correct. So they named them "imaginary numbers" to kind of tell people "don't worry whether this makes sense, it's only used in an intermediate step, they're not real numbers". So, naming a concept after its most common use is exactly what got us into this situation! In addition to the modelling-rotation and solving-cubics aspect of complex numbers, last year I used them in a factoring-things context in a number theory class, for example to prove https://en.wikipedia.org/wiki/Proofs_of_Fermat%27s_theorem_on_sums_of_two_squares https://en.wikipedia.org/wiki/Proofs_of_Fermat%27s_theorem_o... (read the "Dedekind's Proof" section).