  We study complexified Harmonic Oscillator models in two and three dimensions.
Our work is a generalization of the work of Smilga \cite{sm} who initiated the
study of these Crypto-gauge invariant models that can be related to
$PT$-symmetric models. We show that rotational symmetry in higher spatial
dimensions naturally introduces more constraints, (in contrast to \cite{sm}
where one deals with a single constraint), with a much richer constraint
structure. Some common as well as distinct features in the study of the same
Crypto-oscillator in different dimensions are revealed. We also quantize the
two dimensional Crypto-oscillator.
