I had a friend who recently received a job offer. His new employer asked him if he would be willing to skip two weeks notice, and start 5 days later. He agreed and gave notice. His new employer immediately retracted their job offer — they wouldn’t hire someone who would do that. And his old employer wouldn’t give him his job back, for the same reason.
Remember that the most important thing you can walk away with, when leaving an employer, is a good reference. Screw that up badly enough and you’re left with the painful reality that you may be better leaving that job off your resume, thereby creating a job gap (and if you leave it off your job application, a lie).
Two weeks notice is necessary. And I would suggest that if you’re in a management role, it should be at least three weeks.