You sure can feather the propeller without shutting down the turbomachinery - this is commonly done during every shutdown (it takes 30 seconds from feather before the engine may be shut down due to cooling). There are some abnormal procedures as well that call for feathering only, not shutting the engine down inflight.
Originally Posted by CJ Driver
...when the propeller is quite clearly stationary.
A running engine will keep on spinning a feathered prop. On the DH8D, approximately 220 rpm are expected. While the look of a propeller in such a state is certainly different from a normally running example, it is most definitely not stationary.
However, on the incident in question, I strongly doubt the engine was left running.
Firstly, it would take quite a coincidence for a birdstrike to shave off the spinner that cleanly without causing any temporary imbalance. Disintegration of the spinner must come with a few seconds worth of strong vibrations while parts break off unevenly. Also, it is not unlikely that some parts hit the fuselage (in the area of the ice impact trays).
Secondly, the forward 2/3 of the spinner are visible from the flight deck. It does not take a trip to the cabin to find the spinner gone - just turning back and looking out will do the trick; at night, the wing inspection lights will come in handy. So if after a noticeable airframe shudder, the Captain finds the spinner no longer visible from the flight deck, the assumption of serious engine damage, possibly with ingestion of debris, may well be made. This in turn calls for one of the likely best known checklists on this aircraft (from the simulator, I should add): "Engine failure or severe damage after V1" and will result in a full shutdown.
So I think it most unlikely that the colleagues decided to run the engine feathered in such a condition instead of properly securing it.