We study a two-dimensional exactly solvable non-Hermitian PT-non-symmetric quantum model with real spectrum, which is not amenable to separation of variables, by supersymmetrical methods. Here we focus attention on the property of pseudo-Hermiticity, biorthogonal expansion and pseudo-metric operator. To our knowledge this is the first time that pseudo-Hermiticity is realized explicitly for a non-trivial two-dimensional case. It is shown that the Hamiltonian of the model is not diagonalizable.