The question is:
Is the multiplicative group of $\Bbb R$ isomorphic (as a group) to the multiplicative group of $\Bbb Q_p$?
The question is motivated by the observation that the additive group of $\Bbb R$ and $\Bbb Q_p$ are isomorphic (which can be seen from a cardinality argument and both being $\Bbb Q$ vector spaces) and that $\Bbb C_p$ and $\Bbb C$ are isomorphic as fields (apparently a difficult result from class field theory).
Since such an isomorphism need not have any compatibility with addition, so it seems to me that arguments that use sub-rings or ideals are not helpful. On the other hand such an isomorphism would induce representations of $GL(n,\Bbb Q_p)$ to $\Bbb R$ (by composing with the determinant), so maybe a representation theory argument might be the way to go.
Not in general: $\Bbb Q_p^\times$ has $p-1$ torsion elements for odd $p$ while $\Bbb R^\times$ has two. So for $p\notin\{2,3\}$ they are non-isomorphic.
When $p\in\{2,3\}$ one has to work harder. The group of squares in $\Bbb R^\times$ has index two. This is never the case for $\Bbb Q_p^\times$.