Trying to reply to a comment to this answer of mine, I realised I know no better description of the quotient of multiplicative groups $\Bbb Q_p^\times / \Bbb Q^\times$ than just that. Of course I am aware of $$ \Bbb Q_p^\times \simeq p^\Bbb Z \times \mu_{p-1} \times (1+p\Bbb Z_p)$$ for odd $p$, the slight modification for $p=2$, and throwing the logarithm on the last factor, but that does not help a lot. The basic problem being that we mod out a non-closed subgroup.
Well I guess $\Bbb R^\times / \Bbb Q^\times$ or $\Bbb C^\times / \Bbb Q^\times$ are not that easy either, so what should one expect -- but maybe there is something surprising out there?
This question and its answer deal with the corresponding question for additive groups (and admit that there's not much nice to see there).