Solution:

Factoring the polynomial as $(z^{4}+1)(z^{2}+1)=0$, we find that the 6 roots are $e^{ \pm i \pi / 4}$, $e^{ \pm i \pi / 2}$, $e^{ \pm i 3 \pi / 4}$. The calculation then follows from the Law of Cosines or the distance formula.