If you combine Ignoramus idea with mine, I think you will have a god working
system. Use the pressure washer to drive a jet pump (like used on a
waterbed.) The washer would send out a thin, high velocity stream, which
should easily lift a large volume of water 20 ft.

What you described was frequently used in mining to generate large
amounts of mechanical and pumping power. It is called a Pelton Wheel
where you have a large, but very efficient water wheel driven by a hi
pressure jet. You can create this by aiming the pressure washer at the
impeller of almost any pump. You could probably make a swimming pool
pump work if you got access to the impeller. The ultimate result is to
use the power produced by a pelton wheel to drive another pump.