  Continuous, dually epi-translation invariant valuations on the space of
finite-valued convex functions on $\mathbb{C}^n$ that are invariant under the
unitary group are investigated. It is shown that elements belonging to the
dense subspace of smooth valuations admit a unique integral representation in
terms of two families of Monge-Amp\`ere-type operators. In addition, it is
proved that homogeneous valuations are uniquely determined by restrictions to
subspaces of appropriate dimension and that this information is encoded in the
Fourier-Laplace transform of the associated Goodey-Weil distributions. These
results are then used to show that a continuous unitarily invariant valuation
is uniquely determined by its restriction to a certain finite family of
subspaces of $\mathbb{C}^n$.
