Context: I've been reading Tate's thesis, and in it, we defined the character group for $k^{*}$ and $k^{+}$ for a local field $k$. Here we take the range of the characters to be $S_{1}$ for $k^{+}$ and $\mathbb{C}$ for $k^{*}$, where $S_{1}$ is the circle group. It's explained brilliantly in an answer to a previous question I asked about why $S_{1}$ is sufficient for the character group of $k^{+}$ and not for $k^{*}$. This got me thinking, why we don't study characters going to $\mathbb{Q}_{p}$ or $\mathbb{R}$? Studying characters is basically helping us study the group well, right? So doesn't studying these types of continuous homomorphisms $\chi: k^{*} \to \mathbb{Q}_{p}$ give us anything? (I'm asking this in a general setting and not exclusive to Tate's thesis, but some sort of motivation connected to Tate's thesis would be great too).

What do I think: The case about characters to $\frac{\mathbb{R}}{\mathbb{Z}}$ is the same as characters to $S_{1}$, so that question is partially dealt for the case of $\mathbb{R}$. For $\mathbb{Q}_{p}$, we have the totally disconnected topology, so does that make things uninteresting? Or is it because $\mathbb{C}$ is algebraically closed but $\mathbb{R}$ and $\mathbb{Q}_{p}$ are not?