I'm not sure if you can freely use the Q3 engine for a commercial project, Xobim. I recall something being up with that, but I could be wrong.

Anyway, I'm not sure if Epic would even want to sell the rights to Jazz. But if you can look and sound convincing enough (like Stijn said) they may just consider it. On the other hand, it would be a lot simpler to make a spiritual sequel like Strato mentioned.

The main problem with JJ2 was that the publishers simply didn't do their job right, thus a lack of advertising. Even though JJ2 just wasn't as well polished as JJ1, shorter and too easy in general, it probably would have done better if there had been better advertising (for example, in Europe the game did a lot better.) You could perhaps bring that up if you really want to use Jazz.