  An arbitrary form of complex potential perturbation in an oscillator consists
of many exciting questions in open quantum systems. These often provide
valuable insights in a realistic scenario when a quantum system interacts with
external environments. Action renormalization will capture the phase of the
wave functions; hence we construct wave function from Bethe ansatz and
Frobenius methods. The unitary and non-unitary regimes are discussed to connect
with functional calculations. We present a functional renormalization
calculation for a non-hermitian oscillator. A dual space Left-Right formulation
is worked out in functional bosonic variables to derive the flow equation for
scale dependent action. We show equivalence between vertex operator and
permutation operators. The results can be compared with Wentzel Kramers
Brillouin(WKB) calculation. We formally construct the Bosonic coherent states
in the dual space;breaking symmetry will lead to anyonic coherent states. The
limit cycle in renormalization trajectories for complex flow parameters,
especially in extended, complex time limits indicates the need for revisiting
the Bendixson theorem.
