  We study a class of quantum two-dimensional models with complex potentials of
specific form. They can be considered as the generalization of a recently
studied model with quadratic interaction not amenable to conventional
separation of variables. In the present case, the property of shape invariance
provides the equidistant form of the spectrum and the algorithm to construct
eigenfunctions analytically. It is shown that the Hamiltonian is
non-diagonalizable, and the resolution of identity must include also the
corresponding associated functions. In the specific case of anharmonic
second-plus-fourth order interaction, expressions for the wave functions and
associated functions are constructed explicitly for the lowest levels, and the
recursive algorithm to produce higher level wave functions is given.
