just happen to have their partners that are strictly equivalent to real potentials after being exposed to some supersymmetric quantum mechanical treatment [11] or integral, Fourier-like transformation [12]. Jones and Mateo [4] have, moreover, used a Darboux-type similarity transformation and have shown that for the Bender’s and Boettcher’s [1] non-Hermitian PT -symmetric Hamiltonian H = p 2 − g ( ix ) N ; N = 4, there exists an equivalent Hermitian Hamiltonian h = σ − $^{1}$Hσ ; σ = exp ( Q/ 2), where σ is Hermitian and positive definite. Similar proposal was carried out by Bender et al. [3]. For more details the reader is advised to refer to [3,4]. In our current methodical proposal, we try to have our input in this direction and fill this gap partially, at least. Through the forthcoming proposition (in section 2) or through a similarity transformation (in section 3) with a metric operator η (defined in (21) below) we report that for every non-Hermitian complex PT -symmetric Hamiltonian (with positive mass m = m$_{+}$ = + | m | ) there exists a Hermitian partner Hamil- tonian (with negative mass m = m$_{−}$ = − | m | ) in Hilbert space L $^{2}$( R ) = H . In section 3, we also discuss isospectrality and orthonormalization conditions as- sociated with both the Hermitian partner (not necessarily PT -symmetric) and the non-Hermitian PT -symmetric Hamiltonians. An obvious correspondence is constructed, therein. This has not been discussed elsewhere, to the best of our knowledge. We give our concluding remarks in section 4. In connection with an over simplified transformation toy x −→ ± iy ∈ i R ; x, y ∈ R ( x −→ ± iy to be understood as x −→ + iy and/or x −→ − iy ), t’ Hooft and Nobbenhuis [44] have used a complex space-time symmetry transformation 2 A transformation toy: x −→ ± iy ; x, y ∈ R 3 x −→ iy ⇐⇒ p$_{x}$ → − ip$_{y}$ ; x, y ∈ R ,