Combine a conjugate ratio and product

Since $zz^*=1$, $z\ne0$. Multiply the ratio by $z/z=1$:

$\begin{aligned}\frac{z}{z^*}&=-1,\\\frac zz\cdot\frac{z}{z^*}&=-1,\\\frac{z^2}{zz^*}&=-1,\\z^2&=-1,\\z&=\pm i.\end{aligned}$

Both values satisfy the original ratio and product.

From $z/z^*=-1$, obtain $z^*=-z$ (1). Substitute (1) into $zz^*=1$:

$\begin{aligned}z(-z)&=1,\\z^2&=-1,\\z&=\pm i.\end{aligned}$

For $z=i$ or $z=-i$, the conjugate is $-z$, so $z/z^*=-1$ and $zz^*=1$.