Does anyone have a source for a proof on $(\mathbb{C}, +)$ and $(\mathbb{R}, +)$ being isomorphic?

Question

I am looking for a proof for why $\mathbb{C}$ and $\mathbb{R}$ are isomorphic under addition. I understand the outline of the proof; one shows that $\mathbb{R}$ forms a vector space over $\mathbb{Q}$, and $\mathbb{C}$ forms a vector space over $\mathbb{Q}$, and then show that both these have a basis with infinite dimension. However, I would like to see a rigorous proof. Does anyone have a good source for a proof of this?