Isospectral Hermitian counterpart of complex non Hermitian Hamiltonian $p^2-g{x}^4+a\mathrm{/}{x}^2$

In this paper we show that the non-Hermitian Hamiltonians $H=p^2-g{x}^4+a\mathrm{/}{x}^2$ and the conventional Hermitian Hamiltonians $h=p^2+4g{x}^4+bx$ ($a,b\in \mathbb{R}$) are isospectral if $a=(b^2-4g{\mathrm{\hslash }}^2)\mathrm{/}16g$ and $a\ge -{\mathrm{\hslash }}^2\mathrm{/}4$. This new class includes the equivalent non-Hermitian - Hermitian Hamiltonian pair, $p^2-g{x}^4$ and $p^2+4g{x}^4-2\mathrm{\hslash }\sqrt{g}x,$ found by Jones and Mateo six years ago as a special case. When $a=(b^2-4g{\mathrm{\hslash }}^2)\mathrm{/}16g$ and $a<-{\mathrm{\hslash }}^2\mathrm{/}4,$ although $h$ and $H$ are still isospectral, $b$ is complex and $h$ is no longer the Hermitian counterpart of $H$.