I know a proof of this using the fact that both $\mathbb{R}$ and $\mathbb{C}$ are $\mathbb{Q}$-vector spaces of the same dimension and hence are isomorphic.
Also, $$\mathbb{C}/\mathbb{R}\cong \mathbb{R}$$ under the surjective map $a+ib \to a$ from $\mathbb{C}\to\mathbb{R}$.
A more general statement could say that two $\mathbb{Z}$-modules are isomorphic if and only if they are isomorphic as abelian groups. My question is that,
Is there a group theoretic proof of this?
I'm not able to produce a direct proof because they aren't cyclic and they're vector spaces of infinite dimension. Please help.