Difference between i and -i

Consider the two imaginary numbers $i$ and $-i$. Is there any fundamental difference between the two of them? If I take the field $\mathbb{C}$ and apply the map $a + bi \mapsto a – bi$ does the image end up meaningfully different from the field I started with? Or when we write out complex numbers are we arbitrarily choosing which of the non-real solutions to $z^4 = 1$ to call $i$ and which to call $-i$?

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Yes, one of the roots of the polynomial $z^2+1$ is called $i$, the other $-i$; since one is the negative of the other. You can switch the names if you really want to and say that “one root is called $-i$ and the other is called $i$”, but this would not make much difference, right? Traditionally, $i$ is drawn in the upper half-plane and $-i$ in the lower, but this is only a tradition. I am not sure what else would you want to know.

If you have some previous notion of orientation for the plane — some notion of “clockwise” and “counter-clockwise” — then you can specify which solution of $z^2+1=0$ is which. And vice-versa: given a choice of $i$ for $\mathbb C$, you get a corresponding orientation for the plane $\mathbb R^2$ using the usual bijection.