I wonder if it's possible to make a "compressed air" turbine engine
Well, ships have been using turbine engines powered by super-heated water since the late 19th century.  But you still have to have an energy source to heat the steam.  And it's no good for a "jet" engine since the whole point is thrust by accelerating air through the engine, where the turbine and compressor are there just to keep the cycle going.  It would only work for a turbo-prop or turbo-shaft where the turbine's job is to produce torque for doing the work.

Theoretically, you could have a turboprop powered with a steam turbine, or some other heated gas, but you still have to have an energy source.  Maybe some kind of weird and wonderful battery powered water heater, or a tiny nuclear reactor?  Then you also need a huge supply of water or other fluid or gas to heat up. 

It is possible however to have a closed system that recycles the water used for the steam.  You then have pretty much a small nuclear power plant or nuclear sub.  It's theoretically possible though, to have a nuclear powered turboprop that could fly for months.  If it works for subs and aircraft carriers, why not, if you could make the whole thing light enough.

Actually, none of that stuff is really new.  There were nuclear powered aircraft concepts in the 50s, that weren't really practical.

The thing about turbine engines powered by kerosene is that it's still the most efficient way to convert potential energy to kinetic energy in a light weight and trouble free package.  50 years from now?   Who knows.