  Biharmonic maps are the critical points of the bienergy functional and, from
this point of view, generalise harmonic maps. We consider the Hopf map
$\psi:\s^3\to \s^2$ and modify it into a nonharmonic biharmonic map
$\phi:\s^3\to \s^3$. We show $\phi$ to be unstable and estimate its biharmonic
index and nullity. Resolving the spectrum of the vertical Laplacian associated
to the Hopf map, we recover Urakawa's determination of its harmonic index and
nullity.
