I would argue complex numbers don't really have a deeper relation to quantum mechanics. Quantum Mechanics is all about squaring the fact that we must find a rep

I would argue complex numbers don't really have a deeper relation to quantum mechanics. Quantum Mechanics is all about squaring the fact that we must find a representation for a system which simultaneously supports an expression of a probability distribution over measurement outcomes while also providing sufficient representational complexity to support all the symmetries that are found in the kinematic space of the system. Complex numbers happen to serve well for a lot of systems but when you have spinors, for example, you can think of them as just another sort of number the use of which is suggested by the symmetries in (or redundancies in the description of) the underlying system. In other words, the appearance of complex numbers is no more or less mysterious than the allowed transformations of the system itself. At least that is how I've come to understand the question.