Solve |z|=sqrt{x^2-y^2}-1

$$| z | = \sqrt { x ^ { 2 } - y ^ { 2 } } - 1$$

Solve for x (complex solution)

$$x=-\sqrt{\left(Re(z)\right)^{2}+\left(Im(z)\right)^{2}+y^{2}+2\sqrt{\left(Re(z)\right)^{2}+\left(Im(z)\right)^{2}}+1}$$

$$x=\sqrt{\left(Re(z)\right)^{2}+\left(Im(z)\right)^{2}+y^{2}+2\sqrt{\left(Re(z)\right)^{2}+\left(Im(z)\right)^{2}}+1}$$

Solve for y (complex solution)

$$y=-\sqrt{-\left(\sqrt{\left(Re(z)\right)^{2}+\left(Im(z)\right)^{2}}+1\right)^{2}+x^{2}}$$

$$y=\sqrt{-\left(\sqrt{\left(Re(z)\right)^{2}+\left(Im(z)\right)^{2}}+1\right)^{2}+x^{2}}$$

Solve for x

$$\left\{\begin{matrix}x=\sqrt{y^{2}+z^{2}+2z+1}\text{; }x=-\sqrt{y^{2}+z^{2}+2z+1}\text{, }&z\geq 0\\x=\sqrt{y^{2}+z^{2}-2z+1}\text{; }x=-\sqrt{y^{2}+z^{2}-2z+1}\text{, }&z\leq 0\end{matrix}\right.$$

Solve for y

$$\left\{\begin{matrix}y=\sqrt{-\left(z+1\right)^{2}+x^{2}}\text{; }y=-\sqrt{-\left(z+1\right)^{2}+x^{2}}\text{, }&x\neq 0\text{ and }|x|\geq |z+1|\text{ and }\left(z\geq x-1\text{ or }z\geq -x-1\right)\text{ and }z\geq 0\text{ and }\left(z\leq -x-1\text{ or }z\leq x-1\right)\text{ and }\left(z\neq x-1\text{ or }z\neq -x-1\right)\\y=\sqrt{x^{2}-z^{2}+2z-1}\text{; }y=-\sqrt{x^{2}-z^{2}+2z-1}\text{, }&\left(x\neq 0\text{ and }|x|\geq |1-z|\text{ and }z<x+1\text{ and }z\geq 1-x\text{ and }z\leq 0\right)\text{ or }\left(x\neq 0\text{ and }|x|\geq |1-z|\text{ and }z\leq x+1\text{ and }z\leq 0\text{ and }z>1-x\right)\text{ or }\left(x\neq 0\text{ and }x\leq -|1-z|\text{ and }z\leq 0\text{ and }x\leq -\sqrt{x^{2}-z^{2}+2z-1}\text{ and }z\geq x+1\text{ and }z\leq 1-x\right)\end{matrix}\right.$$