The dual group to a compact abelian group is discrete so in particular very much disconnected. I was trying to invent an example of a connected locally compact abelian group with connected dual which is different from $\mathbb{R}^n$, however I failed.
Hence my question:
are there any other examples?
Nice question! In fact there are no other examples.
Let $G$ be a connected locally compact abelian group whose Pontryagin dual $G^{\vee}$ is also connected. Because $G^{\vee}$ is connected, it can have no discrete quotients; taking Pontryagin duals, $G$ can have no compact subgroups. By the Gleason-Yamabe theorem, it follows that $G$ has an open subgroup isomorphic to a Lie group. But since $G$ is connected, the only open subgroup of $G$ is $G$ itself, and hence $G$ is a Lie group.
The connected abelian Lie groups are precisely the products of $\mathbb{R}$ and $S^1$, and since $S^1$ can't occur as a subgroup, it follows that $G \cong \mathbb{R}^n$ for some $n$.