2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/210960We 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.17 pages, Latex, enlarges version, added ref.s., accepted in J.Phys.A, slight alteration in reference section and text, matches journal versionHigh Energy Physics - TheoryMathematical PhysicsQuantum Physicstext