  We show that the elementary modes of the planar harmonic oscillator can be
quantised in the framework of quantum mechanics based on pseudo-hermitian
hamiltonians. These quantised modes are demonstrated to act as dynamical
structures behind a new Jordan - Schwinger realization of the SU(1,1) algebra.
This analysis complements the conventional Jordan - Schwinger construction of
the SU(2) algebra based on hermitian hamiltonians of a doublet of oscillators.
