I think blending third-party concepts with Marvel and DC is the best way to go, that way you have some connection to some of the worlds visited (most realities will heavily feature lore from those two worlds, and it'll be meaningless to your characters if they are not of a reality where at least some of Marvel and/or DC properties exist). 

Also, keep in mind that if your character relies on tools or non-melee weapons, etc., that their supply will be limited. Just the nature of the RP, you won't have a base to go back to and stock back up on supplies. What you have when the group escapes is all you have. Something to consider when brainstorming characters.