1) Bloodwing likes to swoop down to attack you -- so make sure to run (if you need to). I personally didn't have to move around very much when fighting him, but keep it in mind.
2) Secondly, Bloodwing changes elementals several times throughout the battle. To counter this you need to have lots of non-elemental guns OR (instead) a large array of elemental guns so that you can switch to a differenet elemental than what Bloodwing is (if it isn't obvious -- you don't want to attack an elemental enemy with the same elemental that they are using).
3) Aim for his head when possible. This can be hard when he is flying around in the air.
4) If you want to, then you can also get a shield that makes you immune against certain elemental attacks OR you can use an artifact / relic for that (assuming you have one). Many classes also come with skills that can give you more protection against elemental attacks, but that's up to you.
5) If you keep getting lot on health, then I suggest getting a transfusion grenade mod with homing abilities so that it'll attack the bird in the air AND give you health afterward. Some of these mods have elemental damage too, so be careful to use one that doesn't match his current element.
Using Maya the Siren, I destroyed the bird very fast using Phaselock and elemental guns. You can easily solo this creature, duo it, or even kill it in a larger party (though the difficulty level scales).
I've also done it with Gaige the Mechromancer and even Axton the Commando / Commander / Soldier (both Deathtrap and the turret help a good bit -- though the turret to me is a bit better than Death Trap for this battle). People using Salvador the Gunzerker will have a pretty easy time with this boss battle while sporting two guns and if you happen to be using Zer0 / Zero the Assassin, then make sure to take advantage of stealth when needed (or IF needed).
Hopefully this guide gave you some good ideas and tips on killing this monster bird!
Oh...and don't tell Mordecai you have to kill his now slag 'ish pet!
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