We did this in our bog. Can you see the black bowl kind of left of center? t's a two piece bird bath - the bowl is separate from the stand. We drilled a hole in the bowl and ran a tube up the center of the stand and into the bowl. We added a "T" to another line that we have that runs through the bog and attached the tube to that, so water is pushed up from the bog, fills the bowl and spills over the sides. We filled the bowl almost to the top with rocks so the birds have a place to perch and splash away. They love it! Because the water flows constantly it stays clean. It also, oddly, attracts small snails and the birds don't mind the snack while they splash either!
We actually installed it with the intent of growing plants in it, but when we realized how much the birds love it we just let it be.