Suppose $a<0$ and $z_1,z_2,z_3,z_4$ $4$ be the fourth roots of a then $z_1^2+z_2^2+z_3^2+z_4^2$ is equal 

Let $a=-b^4$ where $b>0\text{. }$

Than $z^4=a=b^4(-1)$

  $\begin{array}{l}\Rightarrow z\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=b\left(\pm \cos \frac{\pi }{4}\pm i\sin \frac{\pi }{4}\right)\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\\ \therefore \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{z}_1^2+z_2^2+z_3^2+z_4^2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\\ \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=2b^2\left[\cos \left(\frac{\pi }{2}\right)-i\sin \left(\frac{\pi }{2}\right)+\cos \left(\frac{\pi }{2}\right)+i\sin \left(\frac{\pi }{2}\right)\right]\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\\ \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=0=a+\left|a\right|\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}[\because a<0]\end{array}$