TITLE: Can we construct $\Bbb C$ without first identifying $\Bbb R$?
QUESTION [12 upvotes]: Sometimes it is useful to consider $\Bbb C$ as our primitive and identify $\Bbb R$ as a subset of $\Bbb C$.  Thus we can define $\Bbb R$ (or at least a set with all of the interesting properties of $\Bbb R$) from $\Bbb C$.
This suggests to me that there is some way of constructing $\Bbb C$ without first constructing (or taking as a primitive) $\Bbb R$.  However, I've never seen such a construction of $\Bbb C$ (a quick Google search didn't provide me one, either).  I've the Cayley-Dickson construction and the matrix construction many times, but are they the only known ways of constructing $\Bbb C$?
My question:

Is there a way to construct the set of complex numbers without already having (or first constructing) the real numbers?

REPLY [4 votes]: Any algebraically closed field of characteristic $0$ having the same cardinality as $\mathbb C$ is isomorphic (non-canonically) to $\mathbb C$. This allows one to construct a lot of fields which are abstractly isomorphic to $\mathbb C$ without ever looking at the real numbers. 
For example, the algebraic closure of $\mathbb Q_p$ is isomorphic to $\mathbb C$. However, no isomorphism between them is continuous, so this is not very interesting if you're interested in $\mathbb C$ as a topological field.