$$\begin{aligned} & z^2=-i \bar{z} \\ & \left|z^2\right|=|i \bar{z}| \\ & \left|z^2\right|=|z| \\ & |z|^2-|z|=0 \\ & |z|(|z|-1)=0 \\ & |z|=0 \text { (not acceptable) } \\ & \therefore|z|=1 \\ & \therefore|z|^2=1 \end{aligned}$$