Replacing chemical combustion with electrical heating in a heat engine was an idea that popped into my head a number of years ago while thinking about the heat exchanger for Skylon's SABRE engine. I'll skip the drunken, derailed train of thought that made that particular leap, but off and on since then I've been really intrigued by it, and spent more time thinking about it than I'd care to admit!

First and foremost, I agree that the compressor is probably the most difficult part (though theoretically, if you pre-ionized the gas, you could use magnetic compression). Also, I completely agree that, given current energy densities for electric storage, this is only something that could be useful in some really niche applications.

But that being said, some of those applications are really cool! For example, one of the major challenges of VTOL aircraft is that the rotational inertia of turbines is so great that it's very, very difficult to rotate them during transition from vertical to horizontal flight. Something like this would massively decrease the rotational inertia, making it much simpler mechanically to create tiltwing aircraft.

Also, my understanding is that typically, conventional jet engines are limited primarily by the maximum temperature limit of the turbine blades. Because your compressors here would have to be powered by electricity as well (nothing else makes any sense!), there's absolutely no reason to have a turbine at all; you'd just want a plain old expansion nozzle. That means you could pump way more heat into your plasma, making your engine much more power dense. In other words, your engine could be potentially much smaller for the same thrust, which would be a big deal. Turbine blades need to be both very strong due to their rotational velocity, and extremely temperature resistant because they're literally sitting in the exhaust of a jet engine, which makes them not only really expensive, but also very, very challenging from a metallurgical perspective.

Another, potentially very interesting, application is if you have too little oxygen in your atmosphere to support combustion -- for example, on Mars. Sure, we're about to send a mini electric rotorcraft there, but the atmosphere makes it really very challenging to do that, because the classic "my rotor tips are too close to the speed of sound" problem is much, much more difficult there.

Any kind of ramjet, as you mention, is a possibility, but this would also make it a lot easier to make transition engines (like the J58 that powered the Blackbird) that start as a conventional compressor-fed jet engine and, at cruise speed, transition into a ramjet.

Regardless of application, this is such a fundamental change to the design limitations of jet engines that a lot of the usual design logic simply doesn't apply anymore. Thermodynamics are infamously complicated, which makes it really difficult to draw performance comparisons between an 80-year-old mature technology and something so radically new and different. One way this gets substantially more complicated is that in a traditional jet engine you need to be worried about combustion efficiency, flame stability, etc etc, plus you have to siphon out enough energy to run the engine's compressor, and power the rest of the aircraft (likely indirectly, through an APU!). All of those take a big efficiency hit in traditional engines, whereas this would be, nominally, much better. So my gut would be that, all other things being equal, the powerplant on an airplane with an electric jet engine would be both smaller/lighter and more efficient. But again, this is hard to reason about!

I would be ecstatic to see one of these flying around, but don't expect it to end up in a passenger aircraft any time soon or anything. For that, we need better batteries!