Sounds like something that has happened before and will continue to do so.
Captain heads down trying to find the direct to waypoint. Keep in mind the SIDs out of DOH have lots of closely spaced points and the direct to's there can be on the second/third legs page. Can take longer than usual to find the point and bring it to the top.
Effo decides he will start turn in the approximate direction of the cleared to point. Poor guy banks 25, neglects to keep pitch up (I'm unaware of the trim characteristics on the 787 but the 737 needs manual pitch and/or trim inputs). Transition from peripheral city visual cues off to the right with a turn into a black hole with randomly spaced lit ships off the coast. Down goes the nose.
Where's that point. Ah got it. Heads up and GPWS warning. Whatcha doin' there buddy? Oh crap. I have control!!!
Question is much before the terrain alert there should be a don't sink aural. That should've prevented the supposed 3000fpm and late identification of loss of altitude.