Solve z^2=z^2p^2(1-q^2)

$$z ^ { 2 } = z ^ { 2 } p ^ { 2 } ( 1 - q ^ { 2 } )$$

Solve for p (complex solution)

$$\left\{\begin{matrix}p=-i\left(q^{2}-1\right)^{-\frac{1}{2}}\text{; }p=i\left(q^{2}-1\right)^{-\frac{1}{2}}\text{, }&q\neq -1\text{ and }q\neq 1\\p\in \mathrm{C}\text{, }&z=0\end{matrix}\right.$$

Solve for q (complex solution)

$$\left\{\begin{matrix}q=-\sqrt{1-\frac{1}{p^{2}}}\text{; }q=\sqrt{1-\frac{1}{p^{2}}}\text{, }&p\neq 0\\q\in \mathrm{C}\text{, }&z=0\end{matrix}\right.$$

Solve for p

$$\left\{\begin{matrix}p=\sqrt{-\frac{1}{q^{2}-1}}\text{; }p=-\sqrt{-\frac{1}{q^{2}-1}}\text{, }&|q|<1\\p\in \mathrm{R}\text{, }&z=0\end{matrix}\right.$$

Solve for q

$$\left\{\begin{matrix}q=\frac{\sqrt{p^{2}-1}}{|p|}\text{; }q=-\frac{\sqrt{p^{2}-1}}{|p|}\text{, }&|p|\geq 1\\q\in \mathrm{R}\text{, }&z=0\end{matrix}\right.$$