I really should respec Axton's traits to slag enemies with the turret. Dunno if he can do a lot of soloing that way or not though. Only raid boss I've soloed as Axton is Terramorphus or w/e boss that was in You. Will. Die! (Seriously) was.
What build do you go for? I pretty much go for dealing more damage with non-elementals and upping my explosion damage so I can use DPUH, Cobra, and a Tediore SMG.