Let $\mathbb{K}$ be $\mathbb{R}$ or $\mathbb{C}$. Take $U\subsetneq \mathbb{K}$ open neighborhood of zero verifying:
- $U=U^2 =\{uv\,:\,u,v\in U\}$;
- $U=-U$;
- $\mbox{int}(\mbox{cl}(U))=U$;
- $1\in \mbox{cl}(U)$.
I conjecture that $U$ is actually $U(0,1)$, the open unit ball in $\mathbb{K}$.
Taking $U$ as union of its connected components proves the real case, but for the complex this only leads to the fact that $U$ must be connected (which is nice, but not enough). Also, it's quite easy to see that $U\subset U(0,1)$.
It seems to me that the main problem is that using $4.$ we have some $(a_n)_n\subset U$ converging to one, but we don't control how it converges when working on complex numbers.
EDIT. Clarification: $U^2=U\,U=\{uv\,:\,u,v\in U\}$