I don't understand what it means for something to feel "natural". You can formally define the real numbers in multiple ways which are all isomorphic and coherent. These definitions are usually more complicated than people expect which nicely show that the real set is not a very intuitive object. Same thing for C.

There is not evidence for C. It's a construction. Obviously it shows up in physics models. They are built using mathematical formalism.

If multiple definitions turn out to be isomorphic, that's generally because there is an underlying structure linking the properties together.