  Complex Wadati-type potentials of the form $V(x)=-w^2(x) + iw_x(x)$, where
$w(x)$ is a real-valued function, are known to possess a number of intriguing
features, unusual for generic non-Hermitian potentials. In the present work, we
introduce a class of nonlinear Schr\"odinger-type problems which generalize the
Wadati potentials by assuming that the base function $w(x)$ depends not only on
the transverse spatial coordinate but also on the amplitude of the field.
Several examples of prospective physical relevance are discussed, including
models with the nonlinear dispersion or with the derivative nonlinearity. The
numerical study indicates that the generalized model inherits the remarkable
features of standard Wadati potentials, such as the existence of continuous
soliton families, the possibility of symmetry-breaking bifurcations when the
model obeys the parity-time symmetry, the existence of constant-amplitude
waves, and the eigenvalue quartets in the linear-instability spectra. Our
results deepen the current understanding of the interplay between nonlinearity
and non-Hermiticity and expand the class of systems which enjoy the exceptional
combination of properties unusual for generic dissipative nonlinear models.
