The worst case scenario of personality clash between pilots has to be the Korean Air 2033 incident.
Landing in the rain, the copilot wanted to go-around. The pilot said no. The copilot advanced throttle anyway, then the pilot started swearing and for a moment they were actually fighting over the throttle levers.
The plane ended up touching down, sliding off the runway and was completely destroyed. By amazing good luck, everyone on board survived.
As a pax, I sure as hell want the guys up front to be a 100% operational team.