They are conjugate elements (both roots of the minimal polynomial of C as an extension of R), so they satisfy all the same algebraic properties over R, but they are certainly distinguishable as elements of C.
For example "i" satisfies the polynomial "x-i=0" and "-i" doesn't. It's just that you can't find any such polynomial with real coefficients that differentiates them.
Of course there are lots of non-algebraic ways to distinguish them too. Or did you mean something stronger?
Any true mathematical sentence containing i is still true if you (consistently) replace i with -i. It's why complex numbers are always in the form of a±bi, because there isn't anything that distinguishes a+bi from a-bi other than the conventional sign.
Okay, thanks, I see where you're coming from. That's elementary equivalence, which has some caveats. The truth of sentences that only refer to complex numbers and elementary operations on them is preserved by mapping i to -i. But it's not true if you start involving other structures.
Complex numbers are generally only in that form when obtained as roots of a polynomial. There are lots of applications where different signs have different interpretations. You can say it's a convention, which is true, but that's not quite the same as saying the two signs are the same thing.
For example "i" satisfies the polynomial "x-i=0" and "-i" doesn't. It's just that you can't find any such polynomial with real coefficients that differentiates them.
Of course there are lots of non-algebraic ways to distinguish them too. Or did you mean something stronger?