A field without a canonical square root of $-1$ The following is a question I've been pondering for a while.  I was reminded of it by a recent dicussion on the question How to tell $i$ from $-i$?

Can you find a field that is abstractly isomorphic to $\mathbb{C}$, but that does not have a canonical choice of square root of $-1$?

I mean canonical in the following sense: if you were to hand your field to one thousand mathematicians with the instructions "Pick out the most obvious square root of -1 in this field. Your goal is to make the same choice as most other mathematicians," there should be be a fairly even division of answers.

The splitting field for $x^4+1$ over the reals.
