I looked into this awhile back and found a few pictures on the Internet. Here's a site for a homemade biofilter for a pond.
Basically you treat the pool like an organic pond. It will have tea colored water, but it will be clear. I can't find any of my references but thought you used a filter with activated charcoal, zeolite, and compost (hence the tea colored water). Seems to me you need to use air injection bubblers, too, to keep the water from getting stinky. If water temps get above 80 degrees, you cannot get enough air into the water, so there might be some anerobic bacterial growth to deal with.
Water hyacinth and a few other plants will absorb lots of crud in the water, but they multiply so fast you have to keep on top of them. And the good absorbers are usually considered pests and cannot be disposed of easily. So you have to compost them, which is usually fine unless your pool is also a toxic waste dump. NASA did some research on that way back in the 80s.
I think some of the issues you'll have are
"Black algae" growing on the bottom of the pool
Fear of what harmful bacteria are growing in the pool
Fear of what beneficial bacteria are NOT growing in the pool
Swimming insects (roaches and other swimmers) that will thrive in the pool
Keeping the pool bottom free from debris or letting it go natural.