Force required to move a fluid out of piston at depth I have a cylinder filled with water in it at the bottom of the ocean. Say 100 meters down. In this cylinder is a piston that moves up and down. Its job is to squish the water out the bottom of the cylinder to create an air space in the top half of the cylinder. There are lots of holes to let the water move freely out of the bottom of the piston. There is also plenty of air lines to the surface of the ocean that will let air freely into the top half of the cylinder. 
Assuming we are at sea level and the density of sea water to be 1025 kg/m³, pressure increases by 1 atm with each 10 m of depth.
So the cylinder is under about 10.9204 atm of pressure from the surrounding water at 100 meters down. 
It is my understanding to get the piston to move down and squish the water out of the cylinder you need to overcome the 10.9204 atm of surrounding water pressure. 
My question is this. If you were to push on the ocean bottom piston with another piston of the same displacement located at the surface of the ocean would that not make the 10.9204 atm of pressure irrelevant? (the pistons would we connected with hydraulic lines)
My thinking is the Elevation head due to the fluid's weight, the gravitational force acting on a column of fluid in the hydraulic line from the piston above is already exerting 10.9204 atm of pressure. 
Is this correct or am I missing something?
 A: The pressure of the column of water between the two pistons will cancel out the pressure needed for the one down below to push the water up. So you are correct here.
I will have to correct you about one thing you mentioned in you question though. The pressure you'll have to overcome, should you pump from the lower pump directly is not all of the 10.9204 atm, since you already literally pump air from the surface through the air lines where the pressure is already 1 atm (neglecting the 100 mt air column-you might want to account for that). That pressure aids in your pumping. Even if these lines weren't there you would still be pumping at sea level, so either case you don't account for the atm. pressure.
The forces you will need to really over come when pumping from the surface is the friction in all the lines. 
A: You have to overcome a difference in pressure of 10.00 atm (based on your stated numbers).  Regarding the weight of the hydraulic fluid, this fluid is oil, which is less dense than water.  The pressure at the bottom of your hydraulic line is guaranteed to be lower than the ambient pressure at the bottom of the line, so the liquid head from the column of hydraulic fluid can't do the job that you need done.
Once you put some amount of pressure on the hydraulic lines, such that the pressure at the bottom of the line is equalized with ambient pressure, you STILL have to add additional pressure to allow air into the submerged cylinder.  The additional FORCE that is required equates to the 10 atm difference in pressure multiplied by the cross sectional area of the cylinder that you are pumping air into, where consistent pressure and area units apply.  The pressure in the hydraulic lines must be sufficient to provide this force.
Also note - in my opinion, this is more-or-less a "static flow" situation.  The flow rate of hydraulic fluid will be very low and fluid velocities will be very low as well.  I maintain that friction in the hydraulic lines will be very low, and will actually be the least of your problems in getting air into your submerged cylinder.
