Why is it okay to replace the momentum operator with $p_x + im omega x$?

In this paper the momentum operator was replaced with something that looked very similar to the ladder operator for the non relativistic harmonic oscillator. They started with the Klein Gordon equation and replaced the momentum operator with the operator $p’ = p_x – im omega x$ and replaced the $p_x^2$ term in the KG equation with $frac{1}{2}(p’p’^{dagger} + p’^{dagger}p’)$. The resulting equation reduces to the non relativistic harmonic oscillator equation in the limit $mc^2 gg hbar omega$ (according to the paper) but why is it allowed in the first place to replace $p_x^2$ with that combination of $p’$ and $p’^{dagger}$? The paper they referenced also did this without any explanation.

I noticed that the new operator has the same commutator relation with the position operator, but it’s not Hermitian and as far as I know operators of measurables in QM have to be Hermitian.

Edit: The paper claimed that in the limit $mc^2 gg hbar omega$ the equation
$$left( frac{d^2}{dx^2} – frac{m^2 omega^2 x^2}{hbar^2} + frac{E^2 – m^2c^4}{c^2 hbar^2} right)Psi = 0$$
reduces to the non relativistic harmonic oscillator equation. But after trying to reduce it to that form for a long while, I can’t seem to be able to do it. Now I’m having some doubts regarding the validity of the paper. I’m relatively new to relativistic QM so I can’t really tell.