I have a hippeastrum which is about 4 years old. When the flowers and leaves die, I've been cutting them off and putting them back into the pot on top of the soil. This is in the hope that they rot down and become compost. Is this a good idea or not?
Not really, an indoor plant does not have the same ecosystem in the soil that an outdoor plant does. Hopefully you do not have worms, slugs and snails in the soil of your plant. These are the agents of recycling outside.
You are better off to pick up the dead stuff and apply a dilute fertilizer to replace the nutrients lost.