This work is centred around the role of non-Hermitian Hamiltonians in Physics both at the quantum and classical levels. In our investigations of two-level quantum models we demonstrate the phenomenon of fast transitions developed in the PT-symmetric quantum brachistochrone problem may in fact be attributed to the non- Hermiticity of evolution operator used, rather than to its invariance under PT operation. When it comes to Hilbert spaces of infinite dimension, we show how Lie algebras and Groenewold-Moyal can be explored to construct isospectral Hermitian partners. Alongside,...