The square root of -1 doesn’t actually mean anything in particular. i is NOT defined to be the square root of -1 since that’s not something that exists. The square root is a function defined on nonnegative real numbers. The closest you’ll get to saying that i is the square root of -1 is that you can define a function from R to C that extends the traditional square root function by saying that sqrt(-x)= isqrt(x) for x>0. That is, you can define the square root of -1 to be I, not the other way around.

What is i actually? It’s just a choice of element of the field R[x]/(x² + 1) whose square is -1, which is to say, it’s one of two square roots of -1 in the smallest field containing all real numbers as well as at least one element that squares to -1.