A question on group isomorphism between $(\Bbb{R},+) $ and $(\Bbb{C}, +) $ .

Question

$(\Bbb{R},+) $ and $(\Bbb{C}, +) $ are group-isomorphic.

Consider, two vector space $\Bbb{R}_{\Bbb{Q}}$ and $\Bbb{C}_{\Bbb{Q}}$.

Any two vector space over the same field having same dimension are linearly isomorphic.

Suppose, $T: \Bbb{R}_{\Bbb{Q}}\to \Bbb{C}_{\Bbb{Q}}$ be an Invertible linear map.

And the additive structure are group isomorphic.

Question :

1. Is the above way is correct?

2. I know that $\Bbb{R}_{\Bbb{Q}}$ and $\Bbb{C}_{\Bbb{Q}}$ are infinite dimensional vector space but how to find the cardinality of their hamel basis or how to prove their respective basis have the same cardinality?