Are the groups $\mathbb{C}$ and $\mathbb{R}$ isomorphic? Are the groups $\mathbb{C}$ and $\mathbb{R}$ isomorphic under addition?
And how could I prove this ?
What about $\mathbb{Q}$ and $\mathbb{Q}[i]$ ?

In fact, you can also show that  : $$\frac{\mathbb C^+}{\mathbb R^+}\cong\mathbb R^+$$ by setting the following surjective homomorphism: $$f:\mathbb C^+\to\mathbb R^+,~~~(a+ib)\to a$$ 
