  Valuations on the space of finite-valued convex functions on $\mathbb{C}^n$
that are continuous, dually epi-translation invariant, as well as
$\mathrm{U}(n)$-invariant are completely classified. It is shown that the space
of these valuations decomposes into a direct sum of subspaces defined in terms
of vanishing properties with respect to restrictions to a finite family of
special subspaces of $\mathbb{C}^n$, mirroring the behavior of the hermitian
intrinsic volumes introduced by Bernig and Fu. Unique representations of these
valuations in terms of principal value integrals involving two families of
Monge-Amp\`ere-type operators are established
