So, to actually answer: Complex numbers make it easy to take roots, and using complex numbers, all polynomials with terms up to xN have N roots. These two properties help complex numbers “complete” the real numbers. That is to say; you don’t need to create “super complex numbers” to fix any problems with complex numbers. Here’s the same thing (both equations) using complex numbers:, which is just a direct application of Euler’s equation: . That’s a bit much, but the point is; you can’t get rid of the complex numbers and do QM with just real numbers.