No. Very much no.
The Tab key's behavior is very well-defined in a browser: It tabs you out of the editor box. Going with your feature request would mean killing functionality that everybody expects from their browser. This is not going to happen.
I have created a user script offering the kind of functionality you want (check it out if you're a user script kind of person), and even there I added a way to go back to native tab behavior.
As you'll notice, that way is somewhat awkward. There really isn't a way to make Tab behave in both ways simultaneously, since they obviously contradict each other.
The "press and release Ctrl before pressing Tab" hack (or something along those lines) is okay for someone consciously making the decision to install a user script. But not for the core functionality of a website.