Yes, I know the title seems stupid since the most important property of potential is that it's actually path independent. But I have a point.
I just want to know is it possible to define a function like regular potentials with the difference that they carry the properties of both conservative and non-conservative fields and then derive the well-known ordinary "Potentials" as a special case of the first function which is just path-independent?
The reason I pursue such a thing is there are situations which there exists a non-conservative field in the system and we are studying the system by using Lagrangian or Hamiltonian formalisms. Then of course there wouldn't be such terms as $V$(Potential energy) inside the Lagrangian or Hamiltonian functions while the non-conservative field must somehow show its effects. So there should be a function similar to $V$ which shows the effects of the non-conservative field. Besides I always wondered how can we study non-conservative fields in quantum mechanics since our only chance to show the effect of fields in the "Schrödinger equation" is just manipulating the $V$ term which can only be associated to conservative fields(I know in quantum mechanical levels most of the fields are conservative but I expect quantum mechanics as a universal theory be able to describe every field whether being conservative or non-conservative).
Now the question is:
Is it possible to assign a (possibly path-dependent) function to every field and then derive the regular "Potential" as a special case for conservative fields(of course in order to study non-conservative fields in Hamiltonian and Lagrangian formalisms of classical mechanics or quantum mechanics easier)?