I do see a flaw in the opening on the bottom. The pressure created by the column of water on the right (due to its weight via gravity) will push against the force you are trying to create downward by the air chamber. A 1-way valve (called a "check valve") would reduce the blowback pressure significantly.
Another problem is that the water "entry" (at the top, on the right) is higher than the water "exit" at the left. This means you need more energy to pump the water that way (which is why you are suggesting some way to increase the air pressure in the lower left chamber). It is important to note that there is a distinct physical difference in your setup as opposed to the Heron's fountain setup. Heron's fountain uses 2 separate water containers, with the 2nd already raised so that it has the higher potential energy required <--It is also important to note that this increase in height is a source of energy for the system that does not exist in your setup
My knowledge of chemistry is lacking, but if you plan to use a compound to increase the pressure you will need to make sure it is not water soluble. Also, just how much pressure does this create? (In other words, how much seltzer/whatever will you need and how much will that cost, in comparison to the savings of not having a pump/electricity)
There are a flurry of other problems here, but I will stop there. I hope this doesn't come across the wrong way but perhaps you should look at some introductory physics texts before you spend too much time on your design. This is possible using an energy source other than a pump...but...a perpetual motion machine *IS* impossible. (What you are suggesting is NOT a perpetual motion machine because your source of seltzer/whatever would eventually run out)
Keeping the pressure constant will require a constant source of energy....as you are pumping water up the tube and "out" the top, you are releasing pressure from the system that needs to be reintroduced somehow otherwise the system will simply come to equilibrium.
Also, if the air pressure inside is strong enough to pump the entire column of water, then when water is "falling" from the left side and the check valve opens, it is likely that the air pressure will simply keep the water from falling into the container at all.
EDIT: You will ALWAYS need some other energy source...if you can solve that one, forget the fountain you got yourself a nobel prize in physics.... it comes down to whether any of these alternate energy sources are cheaper than the pump, which I don't believe they are.