[sunshowers]

I'm not a professional, but to me it's clear that whether i and -i are "the same" or "different" is actually quite important.

[impendia]

I'm a professional mathematician and professor.

This is a very interesting question, and a great motivator for Galois theory, kind of like a Zen koan. (e.g. "What is the sound of one hand clapping?")

But the question is inherently imprecise. As soon as you make a precise question out of it, that question can be answered trivially.

[HelloNurse]

Generally, the nth roots of 1 form a cyclic group (with complex multiplication, i.e. rotation by multiples of 2pi/n).

One of the roots is 1, choosing either adjacent one as a privileged group generator means choosing whether to draw the same complex plane clockwise or counterclockwise.

[grumbelbart]

They would never be the same. It's just that everything still works the same if you switch out every i with -i (and thus every -i with i).

[alexey-salmin]

There are ways to build C that result in:

1) Exactly one C

2) Exactly two isomorphic Cs

3) Infinitely many isomorphic Cs

It's not really the question of whether i and -i are the same or not. It's the question of whether this question arises at all and in which form.

[boisterousness]

They're different. Multiplication by i gives a quarter turn counterclockwise. -i gives a quarter turn clockwise.

Opposite quarter turns cancel: (-i)(i) = (-1)(i^2) = +1

Quarter turn twice counterclockwise gives a half turn: (i)(i) = -1

Quarter turn twice clockwise also gives a half turn: (-i)(-i) = -1

[nyeah]

Sure. Either that or the reverse. "They're not the same" in the sense that they can't both be clockwise. "They are the same" in the sense that we could make either one clockwise.

[kergonath]

A bit like +0 and -0? It makes sense in some contexts, and none in others.