We characterize polyharmonic Hopf hypersurfaces with constant principal curvatures as solutions of a fourth-order algebraic equation. We construct six different families of proper polyharmonic hypersurfaces in $ \mathbb{ C P }^n $, and prove that such solutions cannot exist in $ \mathbb{ C H }^n $. Moreover, we study the stability of biharmonic Hopf hypersurfaces with constant principal curvatures in complex space forms.