A topological ring is a ring $R$ which is also a Hausdorff space such that both the addition and the multiplication are continuous as maps.
$F$ is a topological field, if $F$ is a topological ring, and the inversion operation is continuous, when restricted to $F\backslash\{0\}$.
If $F$ is a connected field, then $F$ must be path-connected?
A theorem of Pontryagin asserts that $\mathbb{R}$ and $\mathbb{C}$ are the only fields which are both connected and locally compact Hausdorff (LCH), and maybe we need second-countability as well. More generally, there's a known classification of LCH fields that aren't discrete.
Beyond the LCH case it's hard to even get any control on the underlying abelian group. Apparently there are connected topological fields of arbitrary characteristic, though.