Huggie,
I'm not a diver but seriously looking into doing it in the near future. Yes there is a system you describe. Called a rebreather. Seen it used on TV documentaries where they do not want to disturb what ever it is they are trying to film. It doesn't produce any bubbles. Don't think you can dive for ever though, just a bit longer.
Hopefully someone with more knowledge can fill in any gaps.
Bit of a late reply re re-breathers.
These used to be used by WW2 navy divers, and shortly after by Hans and Lotte Hass in their early diving programmes.
The main estriction with these was the fact that they used a small OXYGEN bottle (not compressed air) as scuba tanks, and used a bag of lime (caustic) to scrub the CO2 from the expelled air. hence no bubbles.
The disadvantages were that going deeper than about 10-12M the oxygen became toxic and could induce seizures in the diver. If that weren't enough, if the lime mixed with water or got wet, it foamed and produced gas (similar to chlorine) and caused severe burns. So definately not for amateurs.
I'm sure there are now more sophisticated versions but would suggest you stick to compressed air. Much safer, and readilly available.
OK lecture over. Enjoy your diving.