Shallow ponds of less than 3 feet of depth often merit wintering fish indoors. Conflicting advice abounds here. Mike White, president of the Midwest Pond Club, notes in his personal experiments that a bubbler run even at the pond's surface still creates a convection current that chills the water a degree or two colder than desired at the pond's bottom. If you do use a bubbler, be sure it is run in the top one-third of the pond's depth to prevent undue chilling of the depths. Pond cleanliness is critical.
Organic sediments decompose with anaerobic bacterial action and produce fish-toxic hydrogen sulfide.
Sedimentary accumulations harbor pathogens and parasites that can attack and harm your fish when their immune systems are essentially dormant. A hole left open in the ice ensures the escape of these toxic gases. Fish are generally safe under a completely frozen pond for up to a week. This allows plenty of time to properly melt through a hole, if necessary. Avoid fish-dangerous sound wave concussions from breaking the ice.