I have an arbitrary complex $2 \times 2$ matrix $B$ and want to find a unitary rotation
$$Q = \begin{bmatrix} c & -\bar{s} \\ s & \bar{c} \\ \end{bmatrix}$$
with $|c|^2+|s|^2 = 1$, such that
$$Q^HB = \begin{bmatrix} f & z_1 \\ 0 & z_2 \\ \end{bmatrix}$$
where $f$ is a real number and $z_1$ and $z_2$ are complex numbers.
This is to triangularize $B$ as it is described in this paper: Computing the Singular Values of 2-by-2 Complex Matrices.
With respect to the notation used in the linked paper, $$ c=\frac{b_{11}}{f}, \quad s=\frac{b_{12}}{f}, \quad f=\sqrt{|b_{11}|^2+|b_{12}|^2}, $$ where $\begin{bmatrix}b_{11}\\b_{12}\end{bmatrix}$ is the first column of $B$. It is trivial to verify that $$ Q^H\begin{bmatrix}b_{11}\\b_{12}\end{bmatrix} =\begin{bmatrix}\overline{c}&\overline{s}\\-s &c\end{bmatrix}\begin{bmatrix}b_{11}\\b_{12}\end{bmatrix}=\begin{bmatrix}f\\0\end{bmatrix}. $$