You can use a regular hang-on power filter--just don't really change the cartridge. If you've kept fish before, like I have, you've probably used the change-the-filter-cartridge-once-a-month rule. Turns out that isn't actually necessary. Yes, the cartridge will look disgusting, but it's still providing mechanical filtration (maybe even a little bit better since there's more gunk inside) and you don't have to lose the nitrifying bacteria living inside of it. If you DO feel the need to change the cartridge, you can always leave part of it, or find a way to add a new cartridge without removing the old one, for at least a few weeks, THEN remove the old one. If your tank has plenty of surface area (plants, lots of hides, sand, large rocks, sponges, etc.), there will be plenty of other places for the bacteria to live and changing the cartridge shouldn't be a huge deal. If you DON'T, then the cartridge is going to be the number one spot for them and removing it could crash your cycle. What stinks is that I literally JUST learned this. I've had fish aquariums my whole life, and we thought we were being good and attentive when we changed the filter cartridges once a month, before they ever got too gross-looking. Most fish are hardier than axolotls and don't seem to be as bothered by less-than-ideal water parameters. With axolotls, as long as the parameters are good, they'll be fine; but high temp, nitrite, and ESPECIALLY ammonia can make them much more prone to infection and can kill them if not corrected.
I've changed my cartridge in my 40-gallon axolotl tank a few times so far, but I think I've gotten away with it (as far as I know, my nitrogen cycle has been pretty good ever since I got it going) since I have a decent amount of surface area elsewhere. I have no substrate, but I have a sponge guard around my filter inlet and a big sea sponge (held down by a rock that I crammed inside it) sitting in one of the bottom corners of the tank.
There are lots of different filters you can use for an axolotl tank (it's much easier just to have one than to try to keep your water clean without one), and the hang-on power filters tend to be the cheapest and easiest. If you already have one, go ahead and use it. If you don't already have one, consider a sponge filter or other types based on the pros and cons.