Let be $\pi : U(2) \to \mathbb{S^3}$ the fibre bundle, defined by $A = (a , b ) \mapsto b$. Does it have a section?

Question

Let be $\pi : U(2) \to \mathbb{S^3}$ the fibre bundle, defined by $A = (a , b ) \mapsto b$, where $U(2)$ is the group of unitary $2 \times 2 $ matrices and $a,b \in \mathbb{C}^2$. ($b \in \mathbb{S}^3$ as $\mathbb{S}^3 \subset \mathbb{R}^4 \cong \mathbb{C}^2$). Does it have a section? Is there a quick way to see that or have a intuition at least?