I always wonder whether he was allowed to go on with his and Hiedeman's research (even in prison).

Attempted murder of the man your doing research for is something that the courts will not look too favorably upon. Even attempted murder carries a pretty healthy sentence. And there's probably no way in the world that he'd be allowed to continue his work in prison. Furthermore the good doctor that survived the attempt wouldn't want anything to do with him.

This episode, when you think of it, was a bit different for Columbo. He initially focused on the death of the nurse, but the final clue involved the attempted murder of Dr Heideman. At its conclusion there was no reference to the initial murder which began the Lieutenants involvement with the case.