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 $-i$?

Solution 1:

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.