Painlevé IV Transcendents Generated from the Complex Oscillator

Supersymmetry transformations are used to generate exactly solvable potentials departing from the complex oscillator. The corresponding Hamiltonians are shown to be ruled by polynomial Heisenberg algebras. A process for reducing the degree of these algebras to 2 is used to connect such systems with the Painleve IV equation, thus leading to a simple algorithm for generating Painleve IV transcendents.