I read a question about ordering of complex numbers, and saw an answer showing that there cannot exist an ordering of the complex numbers because regardless of how $i$ is placed in that order, it would imply that $i^2 = -1$ would be positive.
This proof can obviously be generalized to prove that for any non-zero $a \in Q$ it is impossible for $ai$ to be an element of an ordered field. But there is no obvious generalization of the proof showing that the same hold for all imaginary numbers.
Does there exist a sub-field of the complex numbers containing at least one imaginary number, which can be assigned a consistent ordering?
Or stated in different terms, does there exist a subset of the complex numbers, which is a formally real field and is not a subset of the reals?