same eigenvalues with H$_{x}$$_{;}$$_{m}$$_{+}$ in (16), i.e., where q = ± 1 denotes quasi-parity. Obviously the spectrum remains discrete, real, and bounded from below and the wave functions remain normalizable (cf., e.g., Znoijl [7] for more details), with a c shift of the coordinate up or down. Moreover, the Bender’s and Boettcher’s [1] non-Hermitian PT -symmetric Hamiltonian, with the PT -symmetric potential V ( x ) = − g ( ix ) ν ∈ C ; ν, g ∈ R , ν ≥ 2 , g > 0, would, under the transformation x −→ − iy , yield which is Hermitian (but non- PT -symmetric for odd ν and PT -symmetric for even ν , i.e., conditional PT -symmetric). Whilst Hermiticity is secured in the partner Hermitian Hamiltonian, the boundary conditions and normalizabil- ity are not. Therefore isospectrality is a different issue that remains ”to-be- determined” and to be partially discussed below. The above were just few of the many examples available in the literature where their non-Hermitian PT -symmetric Hamiltonians find Hermitian part- ners in the regular Hilbert space. Whenever one encounters such cases, the possibility of isospectrality should always be tested in the process. In the light of the above proposition, we may observe that our simple transformation toy x ∈ R −→ ± iy ∈ i R , could be interpreted as a counterclockwise/clockwise rota- 8 E$_{n}$ = E$_{m}$$_{+}$ = E$_{m}$$_{−}$ =     2 n + 1; for α = ± 1 / 2 , 4 n + 2 + 2 qα ; otherwise, n = 0 , 1 , · · · , H$_{x}$$_{;}$$_{m}$$_{+}$ = p 2 x 2 m$_{+}$ + V ( x ; m$_{+}$ ) = p 2 x 2 m$_{+}$ − g ( ix ) ν , (19) H$_{y}$$_{;}$$_{m}$$_{−}$ = P 2 y 2 m$_{−}$ + V ( y ; m$_{−}$ ) = P 2 y 2 m$_{−}$ − g ( y ) ν , (20)