If you need the cockpit jumpseat, then my understanding is that you need your passport.
If you are riding on an airline that you code share for, then you wouldn't need a passport as you could non-rev. However, if the flight is full then you would need it for up front.
I wouldn't take a chance and travel without. Also, Toiletduck must be right about needing it to sit in the back of other airlines. I didn't know that you had to have it to sit in back, I thought it was just for CASS (cockpit access).
I always enjoy the gate agents who ask for your license. I give the my cfi and they obviously don't know the difference. Only the CA is required to view licenses, medicals, right? I thought gate agents only looked at company id's and passports?
Anyways, just bring it and you won't have to worry about it.