this is the million dollar question when it comes to ducks!
I'm only a brand newbie when it comes to ducks, but I've been doing a lot of research on filtering very small ponds myself. I have just 2 ducks and want to construct something similar, rather than having to lift up a temporary pond every day to tip the dirty water out. You tend to go through a lot of water, it turns the backyard into mush and I often go away for a week at a time with work, and don't always have someone to change the water, so I really need something that keeps the water reasonably inhabitable until I get back home.
But the filter part isn't easy. Because the ducks poop so much in the water and bring in so many hard foreign objects (like stones), it's very difficult to find a pump that can handle it. I'm really trying to avoid running power from the house (safety and cost reasons) so was looking at a 12v battery with solar cells to recharge. I was looking at the pumps caravans/mobile homes use for their sewerage system. The pumps have an inbuilt grinder that grinds up any waste before it goes through the pump. So it removes the need for the pump to pull through a filtration system (which most pumps don't like), instead can filter after the pump. Seemed like the ideal solution, until I read they aren't designed to run continuous. So you'd have to cycle them intermittently, which I'm not sure yet if it would be enough to keep the water properly filtered. Also boat bilge pumps are another idea, they can handle a fair bit of gunk, but once again they aren't designed to run continuous and usually are fairly high power consumers, which wouldn't suit my battery/solar setup.
The there's the filtration system. Lots of good ideas out there. Once of them I liked was to install a couple of tanks filled with various filtration media like gravel, lava rocks, carbon and even biological filters like plants etc. Pump the water into the tanks and then return back to the pond.
All of these are good in theory, but there doesn't seem to be a lot of success stories. But I'm a gadget man, so I plan on experimenting. I might start with something fairly small scale and see what sort of results I get. At the moment I only have 3 week old ducklings, so I probably need to wait a bit until they get a bit older and start eating coarser food. Their droppings at the moment break down fairly quickly into very small granules (like the food they eat) and sink to the bottom. But I'm assuming they older and bigger they get (along with their change in diet), their dropping will remain more solid and of course larger, which will have a big impact on pump and filtration options..
So sorry I can't give you any answers, but hope it might give you some ideas. I'll share any ideas that I experiment with and the outcome- good and bad!