> Would a ducted fan work better?
This comment is ironic because it initially sounds naive, but is probably the most insightful comment.
You could make a supersonic propeller-driven airplane by mounting the propeller(s) in a shroud, using a powerful engine (possibly geared), and slowing down the airstream to subsonic speed in the shroud (the same way a jet engine is designed.)
The result would be a propeller system that outwardly looks like a jet engine, but internally still has propeller(s) on a shaft(s).
The advantage would be better efficiency at low altitudes if a piston engine was used, since piston engines burn about half as much fuel as jet engines.
Nobody will build this unless high-altitude flight is banned, and thus need the efficiency of piston engines.
Rotax uses essentially a geared lawnmower engine in aviation piston applications that is half the weight of a Lycoming or Continental (because of the gearing.) So hey ... there is room for innovation.
You're not going to get a supersonic helicopter (that looks like a conventional helicopter) without retracting the blades for several reasons, starting with the advancing blade being higher speed than the retreating blade, which makes transsonic and supersonic flight with horizontal blades impractical and risky.
Current fast helicopters have stubby wings, so behave like an airplane, which is kind of cheating. To create a supersonic helicopter would mean either ducted fans instead of blades, or an entirely new technology.
Source: I study aerodynamics.