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?

Any intermediate field between $\Bbb R$ and $\Bbb C$ is in particular an $\Bbb R$-vector subspace of $\Bbb C$. Since $\dim_{\Bbb R}\Bbb C=2$, it is either equal to $\Bbb R$ or to $\Bbb C$.
