Compost and mushroom compost tea can apparently used to grow plants through their entire life cycle. The NPK is a little low, so with amendment, we could make a viable hydroponic solution.
In 5 gallons, I would add 5 pounds of low-sodium content compost, a cup of alfalfa meal, a cup of kelp meal and 1/4 cup rock phosphate and bubble for 48 hours. It would have to be used in a constantly aerated solution.
Check out a couple articles using compost as a base for hydroponic solution: