Scott Manly did an overview of potential ways of propulsion in space including some very wacky setups. The most promising ones seem to be using nuclear or fusion reactions to accelerate tiny amounts of mass to very high speeds. The engineering issues are substantial of course. But definitely goes a bit beyond the limitations of chemical processes (i.e. burning fuels). https://www.youtube.com/watch?v=QEZv_OXA_NI
I'd say beamed energy propulsion is more promising for high speeds. That's because the main problem becomes keeping the vehicle from destroying itself with waste heat, and beamed propulsion keeps the waste heat back at the beam souce. You can even actively cool an object with laser light (via anti-Stokes scattering and the like.)
In the end it's all about a very simple equation: E(kin) = ½mv² so it's all about exhaust velocity. The closer you can get that to its theoretical maximum (e.g. c), the less reaction mass is required.
But the question still remains: is there a practical non-Newtonian way? We already know of rather impractical ones, such has deforming spacetime.
I've wondered if it would be possible to build an efficient rocket engine that accelerates its reaction mass so hard that it gains an appreciable amount of relativistic mass? The fundamental problem with interstellar travel is the amount of reaction mass you need to make the trip in a reasonable amount of time. The rocket equation is a harsh mistress. If you were accelerating each gram of propellant so hard that it gained a metric ton of relativistic mass you could bypass it, assuming you had some sort of lightweight yet near limitless power source.
