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