First, it is the Doom 3 engine, not the Quake 4 engine. It just so happens Quake 4 uses that engine, but it was actually developed for Doom 3.
Second, it has already been suggested and people said it would be a really bad idea.
Third, in another 6 or 7 years (or less if there are some major advancements over the next year or two), Id will probably release the source code for the engine under the GPL.