An involution $\sigma$ is a complex conjugation if not every element of $F^\sigma$ is a sum of squares of elements of $F^\sigma$ and, if $\sigma$ is nontrivial, some element of $F^\sigma$ which is not a sum of squares of elements of $F^\sigma$ is a sum of squares of elements of $F$.
Proof that every complex conjugation satisfies this: $F^\sigma$ has an embedding into the reals. Since every negative real number is not a sum of squares, some element in $F^\sigma$ is not a sum of squares. If $\sigma$ is nontrivial, there is a complex number in $F$ of the form $a+bi$ with $|b|>|a|$. Then $(a+bi)^2 + (a-bi)^2 = 2a^2-2b^2 <0$, so some negative number is a sum of squares in $F$. Of course, negative numbers are not a sum of squares in $F^\sigma$.
Proof that every $\sigma$ that satisfies this is a complex conjugation: $F^\sigma$ is a formally real field, hence it can be given an ordering, and hence an embedding into $\mathbb R$. Moreover, if $x\in F^\sigma$ is an element that is not a sum of squares in $F^\sigma$, then it can be given an ordering such that $x$ is negative. This is because the preordering consisting of the sums of squares can be extended to a preordering including $x$, and thus by Zorn's lemma to an ordering of the field. For $x\in F^\sigma$ an element which is not a sum of squares in $F^\sigma$ but is a sum of squares in $F$, applying this construction gives a real embedding of $F^\sigma$ that extends to a complex embedding of $F$.