  We provide walks through dimensions for isotropic positive definite functions
defined over complex spheres. We show that the analogues of Mont\'ee and
Descente operators as proposed by Beatson and zu Castell [J. Approx. Theory 221
(2017), 22-37] on the basis of the original Matheron operator [Les variables
r\'egionalis\'ees et leur estimation, Masson, Paris, 1965], allow for similar
walks through dimensions. We show that the Mont\'ee operators also preserve, up
to a constant, strict positive definiteness. For the Descente operators, we
show that strict positive definiteness is preserved under some additional
conditions, but we provide counterexamples showing that this is not true in
general. We also provide a list of parametric families of (strictly) positive
definite functions over complex spheres, which are important for several
applications.
