In my search online for a fantastical air propulsion system, I came across the pump-jet used in some watercraft. I was wondering—aside from a jet engine, is there an air-based equivalent that functions the same way, and if not, could one still be created and feasible work? I guess what I mean is, is it possible to design a method of air propulsion that is basically the same as the water-based pump-jet but technically can’t be considered contemporary turbofan or turbojet.

There exist... But not for avionic appliances, in the industry there exist somethimg called pneumatic motor and is used in the meal industry

7481716
generally, air based versions are turbo jets because you already need to power a fan, might as well make the combustion happen within the same system. Without it, you just have fans or pumps blowing air. The closest thing is a ducted fan, which can be an electrically powered axial fan.

Otherwise, you could have a piston engine power the fan, but since aircraft generally operate in steady state, something turbine engines really excel at, you're going to just make a worse propulsion system.

I believe the S-71 blackbird used something similar to a pump jet and regular jet combo SR-71 Blackbird engine