Let $A$ be a real matrix \begin{bmatrix} w & x \\ y & z \end{bmatrix} with complex eigenvalues $a+bi$ and $a-bi$. We're looking for an algorithm to find the Givens rotation matrix \begin{bmatrix} c & s \\ -s & c \end{bmatrix}
such that we have
$$ A = \left[ \begin{array}{cc} c & s \\ -s & c \end{array} \right] \left[ \begin{array}{cc} a & b_1 \\ b_2 & a \end{array} \right] \left[ \begin{array}{cc} c & -s \\ s & c \end{array} \right] $$
where $b_1b_2 = -b^2$