Solution: Factor $ z^4 + z^2 + 1 = (z^2 + z + 1)(z^2 - z + 1) = 0 $. Solving $ z^2 + z + 1 = 0 $ gives roots $ z = \frac{-1 \pm i\sqrt3}2 $, with imaginary parts $ \pm \frac{\sqrt3}2 $. Solving $ z^2 - z + 1 = 0 $ gives $ z = \frac{1 \pm i\sqrt3}2 $, with imaginary parts $ \pm \frac{\sqrt3}2 $. The maximum imaginary part is $ \frac{\sqrt3}2 = \sin 60^\circ $. - ToelettAPP