My preference for shallow tanks is to use no filter. For some, I include an airstone.
There are small submersible filters that will work in only about 7-8cm of water, but there are a couple of problems with using them. One is that they usually generate a significant current. This can be overcome by placing them behind rocks or something like that. The second problem is that they generate heat. In summer, when we're all struggling to keep our tanks cool, this can be counterproductive.
Keep in mind that filters do not really "clean" the water at all. What they do is to move all the crud into one place, rather than having it all over. The stuff that builds up in the water can only be cleaned by removing water from the tank. My philosophy is that if I have to remove water from the tank frequently, I'll just remove the dirt at the same time, so there's not much point in having a filter.
I think if you google "shallow submersible aquarium filter" you'll find photos of suitable filters. But you might also want to consider continuing w/o a filter or perhaps just adding some aeration.