Does there exist a field $K$ such that $\mathbb R \subsetneq K \subsetneq \mathbb C$? I'm thinking of unions of $\mathbb R$ with some subset of $\mathbb C$ but am not sure how to approach this without ending up with all of $\mathbb C$. Doe anyone have any suggestions?

and just very simply if $z \in \mathbb{K} \subset \mathbb{C}$ but $z \notin \mathbb{R}$ then it must be of the form $a + ib$ with $a,b \in \mathbb{R}$ and $b \ne 0$. hence $i \in \mathbb{K}$ so $\mathbb{C} \subset \mathbb{R}(i) \subset \mathbb{K}$
