  We find the spectra and eigenfunctions of both ordinary and supersymmetric
quantum-mechanical models describing the motion of a charged particle over the
$\mathbb{CP}^{n-1}$ manifold in the presence of a background monopole-like
gauge field. The states form degenerate $SU(n)$ multiplets and their wave
functions acquire a very simple form being expressed via homogeneous
coordinates. Their relationship to multidimensional orthogonal polynomials of a
special kind is discussed. By the well-known isomorphism between the twisted
Dolbeault and Dirac complexes, our construction also gives the eigenfunctions
and eigenvalues of the Dirac operator on complex projective spaces in a
monopole background.
