> Is this the shadow of something natural that we just couldn't see, or just a convenience?

They originally arose as tool, but complex numbers are fundamental to quantum physics. The wave function is complex, the Schrödinger equation does not make sense without them. They are the best description of reality we have.

The schroedinger equation could be rewritten as two coupled equations without the need for complex numbers. Complex numbers just simplify things and "beautify it", but there is nothing "fundamental" about it, its just representation.