I wonder if one can place mathematical concepts on a spectrum from discovery to invention? To me, the pythagorean theorem feels much more like a discovery of a "hidden" eternal truth that was once beyond our grasp. But to me, complex numbers seem more like a notational choice, more akin to an invention. To put it another way, I would expect aliens to have the pythagorean theorem, but perhaps different notation for complex numbers, like representing them with matrices and computing over them with linear algebra. Such a representation would reduce the distinction of the "imaginary axis" which is really not that special.

I may be naive; I haven't worked or thought with complex numbers since completing my basic education.