Think about getting a second pump. Continue to use the one you have, to run the pretty waterfall and aerate the water.
Then you could use the other to operate a Tetra non-pressurized bio filter. About the size of a 5-gallon bucket. I had one for years and it worked great. It is not pressurized, though, so that means the water must feed by gravity into your pond, and so the hose going to the pond has to be down below the surface. That's why you still need your original set up to splash into the pond and add oxygen to the water.
I recently saw a set-up similar to what Kat described. Simply some filter media set at the edge of the spillway, so it caught gunk before it went back into the pond. That might work OK for a few goldfish, but if you move to koi, or overcrowd your pond (as most of us do), you will need a real filter.