Given the $n+1$-dimensional complex vector space together with its standard hermitian scalar product and its standard basis $(\mathbf{e}_i)^j = \delta_i^j$. Given a vector $v$, is there a general way to obtain a matrix $R \in U(n+1)$ such that $$ R \mathbf{e}_{n+1} \in \mathbb{C} v $$ that is, $R$ rotates $\mathbf{e}_{n+1}$ into the ray of $v$? There are many such $R$, but it would suffice to find one (the others a found through $U(n)$ transformations) however it should depend smoothly on the ray.
for $n=1$, this can be done as follows:
A general $\text{SU}_2$ transformation has the form \begin{align*} R = \begin{pmatrix} a^* && b\\ -b^* && a \end{pmatrix} \ , \end{align*} where \begin{align} |a|^2 + |b|^2 = 1 \ . \label{eq:abConstraint} \end{align} Demanding that $R \mathbf{e}_2 \in \mathbb{C}z$ leads to the following equation, with $\lambda \in \mathbb{C}^\times$: \begin{align*} (b,a) = \lambda (z^1,z^2) \ . \end{align*} To obtain more explicit formulas, restrict first to $z^1 \neq 0$. Then: \begin{align*} \frac{z^2}{z^1} = \frac{a}{b} \ . \end{align*} from there one derives, using the constraint on $a$ and $b$, $|b| = (1 + |z^2/z^1|^2)^{-1/2}$ and $|a| = |z^2/z^1||b|$. Hence denoting the phases of $a,b$ by $\theta_a,\theta_b$ respectively, \begin{align} \frac{z^2}{z^1} = \left|\frac{z^2}{z^1}\right| e^{i(\theta_a-\theta_b)} \ . \label{eq:abPhases} \end{align} This equation fixes the relative phase of $a,b$. However, the requirement of smoothness of $R$ fixes $\theta_b = 0$ (or a constant). The reason is that the vanishing of $z^2$ is allowed. In this case the phase of $z^2/z^1$ becomes ill-defined. So if this phase appears in $R$, its coefficient should go to zero as well in this case. This fixes \begin{align*} a_1 = \frac{\frac{z^2}{z^1}}{\sqrt{1 + \left|\frac{z^2}{z^1}\right|^2}} \ , \quad b_1 = \frac{1}{\sqrt{1 + \left|\frac{z^2}{z^1}\right|^2}} \ . \end{align*}