The discussion following Mike Shulman's 2013 post From Set Theory to Type Theory was useful to me to see what to say about this situation type-theoretically while writing section 3.4.3 of my book Modal Homotopy Type Theory.
Mike distinguishes between the introduction of the type of complex numbers as a particular type and as an abstract type. As a particular ...
Could this be the question you mean? It's not exactly about $i$ versus $-i$, but it mentions it as a motivating example.
"Co-ordinate-free" mathematics for general structures?
Even if it isn't, don't miss the reference there to Shapiro - Identity, indiscernability, and ante rem structuralism: The tale of $i$ and $-i$.
This question relates directly to issues that are often discussed in the philosophy of mathematics.
According to one of the standard accounts of structuralism, what mathematical objects are at bottom are the structural roles that they play within a mathematical system. One might define the equivalence relation whereby element $a$ in structure $A$ is ...