ideally for axolotls a two stage filter is required.. first stage = mechanical, this tends to be foam/sponge which is used to trap debris/waste, second stage = biological, this tends to be media such as ceramic rings, lava rock etc.. the biological stage removes ammonia and nitrites and houses the bacteria colony, some filters have a third stage which is chemical, this tends to be media such as activated carbon, zeolite, ion-resin, chemical filtration tends to need to be changed/recharged monthly, chemical filtration is used to remove chemicals/medications and at times nitrogen compounds.
external canister filters tend to be the best option but is also the most expensive.
with axolotls they don't mind water flow but don't like water pressure, use a spray bar or a means of defusing the water.
use sponges/guards over water inlet to prevent injury.
due to lack of substrate etc.. in an axolotl tank always use a filter for double tank capacity.