i believe i understand what you meant. Your left hand keeps playing One note octave key. For example, you play low C with your pinky and you play the C an octave higher with your thumb. I always find myself doing that all the time.
If you are playing the piano, it is more difficult to invent something to do with your left hand. But you can try playing bass lines with your left hand while your right hand plays chords, or your left hand can play chords while your right hand plays melody (which is very rare).
Or left and right hand can join up chords together (make some ninth chords or thirteenth chords).
But if you are playing the organ, it is better. You can just learn how to play the bass pedal, and just play some chords with your left and right hand.