Non-hermitian Hamiltonians and the Painlevé IV equation with real parameters

► We use higher-order supersymmetric quantum mechanics to derive deformed algebras. ► We obtain complex solutions of the Painlevé IV equation with real parameters. ► We analyze the algebraic structure of the corresponding non-hermitian Hamiltonians. ► We study the eigenfunctions and the energy spectra of non-hermitian Hamiltonians.