The theory is good, but the scale is all wrong. Jet engines consume truly huge quantities of air and an AL 80, or even several AL80's would not provide a whole lot of range as it is just not that efficient blowing out a nozzle at the low speeds involved.
To be more effective you'd have to have a rocket using air, nitrox or O2 as an oxidizer and burning some other type of fuel. Hydrogen would be preferable as the by products would essentially be water and would not overly alarm the environmentalists (more than they would already be alarmed at the idea of rocketing around the reef.)
Using the compressed air to run a propeller would be more efficient and has I think been tried before. Using CO2 would be an even better idea as it would allow much more gas in an AL80 sized tank.
Hmmm...there could be a workable idea there. CO2 powered motors used to be pretty popular for small freeflight aircraft models. They are simple and lightweight and no combustion would be involved other than lightly carbonating an already heavily calcium carbonated reef, there would be no ill environmental effects. A very simple recoil starter would be a safer alternative to spinning the blades maually to get the prop and motor moving and the motor would then keep running until it ran out of CO2 or until you shut off the CO2 to the engine.
All you'd need is a CO2 tank, a simple regulator, a control/throttle valve, a recoil starter and an massively upscaled single cylinder model aircraft CO2 motor driving a gear box and propeller and a housing to hold it all in a streamlined scooter shaped package. You'd probably need a check valve in the exhaust to prevent sea water from backing up into the engine when it was not in operation but there would be much less that would need to be sealed and waterproofed compared to an electric scooter and it would probably be more reliable than an electric scooter. Plus you could "stage" other CO2 tanks and change them on long dives, unlike batteries in electric scooters.
Anybody know a patent attorney and someone with some money to invest?