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 $\mathbb{Q}[i]$ ?

The groups in question are all additive groups of fields of characteristic zero.  In general the additive group $(F,+)$ of a field of characteristic zero is a uniquely divisible group: for all positive integers $n$, the map $[n]: F \rightarrow F$ given by $x \mapsto nx$ is an isomorphism.  Indeed, it's a homomorphism for any commutative group and its inverse is $x \mapsto \left(\frac{1}{n}\right) x$.
I claim that any uniquely divisible commutative group $M$ admits the structure of a $\mathbb{Q}$-vector space in a unique way.  For any nonzero rational number $\frac{p}{q}$ and any $x \in M$, we must define $\frac{p}{q} x$ to be the unique 
element $y$ of $M$ such that $qy = px$.  It is easy to check that this works.
The only invariant of a vector space $V$ over any field $K$ is its dimension.  Further, when the cardinality of $V$ is greater than the cardinality of $K$, the dimension of $V$ is equal to the cardinality of $V$.  Thus:
$\mathbb{R}$ and $\mathbb{C}$ are both $\mathbb{Q}$-vector spaces of continuum cardinality; since $\mathbb{Q}$ is countable, they must have continuum dimension.  Therefore their additive groups are isomorphic.  
$\mathbb{Q}$ is a one-dimensional $\mathbb{Q}$-vector space whereas $\mathbb{Q}[i]$ is a two-dimensional $\mathbb{Q}$-vector space, so their additive groups are not isomorphic.  (Note that the dimension of a $\mathbb{Q}$-vector space is the maximum cardinality of a $\mathbb{Z}$-linearly independent set.  This leas to GEdgar's answer.)
