So first off, why Quake 3? After all, CopperLicht has already been there and done that, and I think many people won't recognize this as much of a step up from the Quake 2 demo I already did. Not to mention that along with my Doom 3 model loader this is my 3rd id-related demo in a row, which may seem a bit boring to some people.
For me it mostly comes down to what I wanted to achieve personally. Thus far each of my demos have had been the result of a specific goal on my part: My Spore demo was just to familiarize myself with WebGL. The Hellknight model was to toy around with animation and game-oriented file formats. The Quake 2 demo was a (somewhat failed) stab at large scenes that the user could navigate. I had intended next to move on to some special effects (I still want to try Megatexture-style streaming, for example), but the problems with the Quake 2 demo left me itching for some closure. So I redirected my efforts and decided that I wanted to do a project that created a "real world" game environment in WebGL. Something that looked and acted like an actual game, running at framerates that would actually be considered playable, without "dumbing it down" for the browser. Partially for the challenge, and partially to demonstrate that yes, WebGL is (or at least will be when released fully) a viable platform for game development.