Let $\pi : \mathbb{R} \to \mathbb{R}/ \mathbb{Q}$ be the canonical projection. With the axiom of choice we "know" that there are choice functions $\alpha : \mathbb{R}/ \mathbb{Q} \to \mathbb{R}$ with $\pi \circ \alpha = id_{\mathbb{R}/ \mathbb{Q}}$ (i.e. with $\alpha(x) \in x$ for all $x \in \mathbb{R}/ \mathbb{Q}$).
Are there such choice functions which are group homomorphisms (with respect to addition)?
$\mathbb{R}/\mathbb{Q}$ is a $\mathbb{Q}$-vector space. Choose a $\mathbb{Q}$-basis for $\mathbb{R}/\mathbb{Q}$, and pick a preimage for each basis vector in $\mathbb{R}$. By mapping each basis vector to the chosen preimage, you have successfully constructed a $\mathbb{Q}$-linear map $\mathbb{R}/\mathbb{Q} \to \mathbb{R}$ that splits the quotient map. This is in particular an additive homomorphism.
More generally, any short exact sequence $$0 \to U \to V \to W \to 0 $$ of vector spaces splits.