jmc

Since $|z_1|=|z_2|=1,$ $|z_1{z}_2|=1.$ Let $$z_1{z}_2=si,$$where $s\in \{-1,1\}.$

Similarly, $\left|\frac{z_1}{z_2}\right|=1.$ Let $$\frac{z_1}{z_2}=t,$$where $t\in \{-1,1\}.$

Multiplying these equations, we get $z_1^2=sti.$ This equation has two solutions.

Thus, there are two choices of $s,$ two choices of $t,$ and two choices of $z_1,$ giving us $\boxed{8}$ possible pairs $(z_1,z_2).$