The authors analyze a class of non-Hermitian quadratic Hamiltonians, which are of the form \( H = {\mathcal{A}}^{\dagger} {\mathcal{A}} + \alpha {\mathcal{A}} ^2 + \beta {\mathcal{A}}^{\dagger 2} \), where \( \alpha, \beta \) are real constants, with \( \alpha \neq \beta \), and \({\mathcal{A}}^{\dagger}={d\over dx}+W(x)\) and \({\mathcal{A}}=-{d\over dx}+W(x)\) are generalized creation and annihilation operators. These operators are non-Hermitian for \(\alpha \neq \beta\). It is shown that the eigenenergies are real for a certain range of values of the parameters. A similarity transformation \(\rho\), mapping the non-Hermitian Hamiltonian \(H\) to a Hermitian one \(h\), is obtained. It is shown that \(H\) and \(h\) share identical energies. Two explicit examples are considered in this work, namely, models based on the trigonometric Rosen-Morse I (\(W(x)=-A_1\cot x - {B_1 \over A_1}, A_1>0, B_1>0\)) and the hyperbolic Rosen-Morse II (\(W(x)=-A_2\tanh x - {B_2 \over A_2}, A_2>0, B_2>0, B_2<A_2^2\)) type potentials.
Moreover, there is discussion of the case when the non-Hermitian Hamiltonian is \({\mathcal{PT}}\) symmetric, i.e. \(H\neq H^{\dagger}, H{\mathcal PT}={\mathcal PT}H\), where \({\mathcal P}\) stands for parity and \({\mathcal T}\) denotes time reversal operators: \({\mathcal P}\psi(x)=\psi(-x)\), \({\mathcal T}\) is complex conjugation.