Any CONSTANT companion I brought with me would probably get a bit grating after a couple centuries. So a better deal would be to bring someone to focus on getting the human race on a better footing out of the gate while I go around seeing the sites, meeting people and molding the world in my image.
So, a doctor, or scientist of some sort, who can teach folks about hygiene and medicine and some fundamental truths about the universe. Then we make a bee-line to Greece, where I carve out civilization as a warlord and my buddy sets up shop in the capital. It'd be really, really interesting to see how humanity turns out with someone at the helm focused on lasting five thousand years.
You know, Moira from Fallout 3 would probably be all OVER this nonsense. And would be perfectly happy staying put somewhere while someone runs around the globe getting stuff done.