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 are introducing a specific form of regularization which does not break the bare model symmetry to analyze vacuum energy the gap in the system. We compare the results with methods like 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 indicating the need for revisiting the Bendixson theorem.