Excalibur,
I am also just a novice regarding ponds and filters and bio-converters. However, after all the reading and researching I have done thus far, coupled with my personal experiences and what I have observed in nature, I totally agree with Waterbug's information related to this subject.
I sincerely believe that bakki shower and trickle tower type bio-conversion or bio-filters are the best all around systems for backyard ponds.
I also investigated moving bed systems, expressly something incorporating Kaldness K1 or K3 media, but I have become less enthused with these systems for small operations and ponds because I believe that they demand much more energy to "fluidize" the media bed.... Which means in simpler terms that it requires more energy to keep the media in motion and bumping into itself continuously within a bath of water. That is just common sense regarding the energy expended. Pumping the water up about the same head height and just allowing it to trickle down through stationary media seems more energy efficient than trying to pump enough water and pump enough O2 into the water and agitate the water sufficiently to make a bunch of friction prone media discs keep rolling around like a washing machine. And then, I am not so certain that it is AS effective in the first place. I am thinking not.
In applications, such as bakki showers, I envision small diameter ladies hair curlers, the very open and porous type... (breathable) as being about the best media. That is my opinion based upon what principles are involved. You need something with a lot of "open" surface area. Plastic bottle brushes come to mind, also. Matala mats and rolls are probably very good, but they are more expensive. I have bought several rolls, but haven't yet put them to use. I will use them, but I will apply them in an open air "trickle shower" scheme. I think this will work. I would not apply them in a submerged soaking fashion, like a skippy filter system, though. I think they would be better applied in a downflow shower.
My pond is a big ass fishbait tank. It is mainly for my fishing bait. I have dirty, rough fish and sometimes 30 pound snapping turtles and 6 to 60 pound catfish in my "pond". Sometimes, when I catch my baitfish, I dump the whole 5 gallon bucket in the tank with leaves, sticks, mud and algae and who knows what, it all goes right in. It is pretty crude quite often. So, you can understand that I need to do some "special" filtering compared to someone's quaint little patio pond. I've got a "REDNECK" pond! LOL
I have to manage this system differently than most of your projects. So I have been devising and developing my own DIY filtration and water conditioning system for quite a while now. I am close to having a decent system engineered. I have had a few fair examples working before, but... now with the information that I have gained from this forum, I have refined and improved upon those old designs, and I currently have an entirely new system under construction and working on it almost daily.
Catfishnut