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We derive a one-step extension of the well known Swanson oscillator thatdescribes a specific type of pseudo-Hermitian quadratic Hamiltonian connectedto an extended harmonic oscillator model. Our analysis is based on the use ofthe techniques of supersymmetric quantum mechanics and address variousrepresentations of the ladder operators starting from a seed solution of theharmonic oscillator given in terms of a pseudo-Hermite polynomial. The role ofthe resulting chain of Hamiltonians related via similarity transformation isthen exploited. In the second part we write down a two dimensionalgeneralization of the Swanson Hamiltonian and establish superintegrability ofsuch a system.

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