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.

I don't feel that the question is really a coherent one.  Let me try to illustrate that as follows.
Suppose I hand you the field $\overline{\mathbb{C}} = \{a - bi \ | \ a+bi \in \mathbb{C} \}$. It seems that in your sense of canonical, the canonical square root of $-1$ in this field is $-i$.  
But wait: $\overline{\mathbb{C}}$ is not a(n even set-theoretically) different field from $\mathbb{C}$!  It is just being presented differently: namely starting with the incarnation you usually have of $\mathbb{C}$ (which you have not specified in your question, but is rather implicit in its formulation that you have one) and compose it with the complex conjugation automorphism.  The point here is that it doesn't make sense to distinguish between the fields $\mathbb{C}$ and $\overline{\mathbb{C}}$, only to distinguish between different identifications of these fields.  
In summary, whenever your field has an automorphism which moves an element $x$ such that $x^2 = -1$, there is no canonical choice between $x$ and $-x$, at least not in any mathematically robust sense.  
