Solve p-q+1/p+q-1=q/i-p

$$\frac { p - q + 1 } { p + q - 1 } = \frac { q } { i - p }$$

Solve for p

$$\left\{\begin{matrix}\\p=-\frac{\sqrt{-\left(2q+\left(-1+i\right)\right)^{2}}}{2}+\left(-\frac{1}{2}+\frac{1}{2}i\right)\text{, }&\text{unconditionally}\\p=\frac{\sqrt{-\left(2q+\left(-1+i\right)\right)^{2}}}{2}+\left(-\frac{1}{2}+\frac{1}{2}i\right)\text{, }&q\neq 1-i\text{ and }q\neq 1\text{ and }q\neq 0\end{matrix}\right.$$

Solve for q

$$\left\{\begin{matrix}q=\frac{\sqrt{-\left(2p+\left(1-i\right)\right)^{2}}}{2}+\left(\frac{1}{2}-\frac{1}{2}i\right)\text{, }&p\neq i\text{ and }p\neq 0\\q=-\frac{\sqrt{-\left(2p+\left(1-i\right)\right)^{2}}}{2}+\left(\frac{1}{2}-\frac{1}{2}i\right)\text{, }&p\neq i\end{matrix}\right.$$