New 1-step extension of the Swanson oscillator and superintegrability of its two-dimensional generalization

•We show that the Swanson system admits a 1-step rational extension.•We provide a flow-chart linking the Swanson model to its generalized counterpart.•We examine superintegrability of the Hermitian equivalence of the Swanson oscillator.

We derive a one-step extension of the well known Swanson oscillator that describes a specific type of pseudo-Hermitian quadratic Hamiltonian connected to an extended harmonic oscillator model. Our analysis is based on the use of the techniques of supersymmetric quantum mechanics and addresses various representations of the ladder operators starting from a seed solution of the harmonic oscillator expressed in terms of a pseudo-Hermite polynomial. The role of the resulting chain of Hamiltonians related to similarity transformation is then exploited. In the second part we write down a two dimensional generalization of the Swanson Hamiltonian and establish superintegrability of such a system.