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.