We know that $\mathbb{R}$ and $\mathbb{Q}$ have a unique structure as ordered fields with the usual order, and that $\mathbb{C}$ cannot be realised as an ordered field. Various non-trivial subfields of $\mathbb{R}$, such as $\mathbb{Q}(\sqrt 2)$, have multiple orderings, but I can't yet completely rule out the possibility of there being other subfields of $\mathbb{C}$ with unique order structure.
Clearly any such subfield cannot contain $bi$ for any $b\in\mathbb{R}^\times$. How can I determine whether there are any other possibilities than $\mathbb{R}$ and $\mathbb{Q}$?