Your view of the complex numbers is the rigid one. Now suppose you are given a set with two binary operations defined in such a way that the operations behave well with each other. That is you have a ring. Suppose that by some process you are able to conclude that your ring is algebraically equivalent to the complex numbers. How do you know which of your elements in your ring is “i”? There will be two elements that behave like “i” in all algebraic aspects. So you can’t say that this one is “i” and this one is “-i” in a non arbitrary fashion.