Or… You can have two Zero’s! One for Co-op and one for solo. Then you can have a melee Zero and a Sniper Zero for Co-op, and same for solo. And then… You get the idea, don’t you! 
Gaige takes some figuring out before you really start appreciating her. But once that happens, you can have a lot of fun with her. I recently built a Gaige for Melee. It’s something that I can really recommend. At level 20, you should be getting to the “Rakkaholics Anonymous mission” pretty soon. If you turn the mission in to Moxxi, whe will give you the Rubi. Combine this gun with a nova shield and put 5/5 in Blood Soaked Shields, and you’re ready to rock.
An Eviscerating Rubi will do more damage, but any version will do for now. Since it’s melee and since it heals you, the level of it doesn’t matter. You can also go with a roid shield for more melee damage, but then you’d miss out on all the wonderful novas.
Eventually, this build works at it’s peak with the Hide of Terramorphous and an Eviscerating Grog Nozzle. Close distance to enemy while slagging - Shield strip - Nova - Massive melee bonus - Kill - BSS kicks in with full shields… If you also have Sharing is Caring, you and Deathtrap can both beat up on the same enemy at the same time. Life’s beautiful! 
This is also a VERY powerful build for mobbing, btw!