Having once met someone very similar to me, I thought they were kind of big headed, opinionated and a real pillock to be around. If my parallel universe counterpart was exactly like me, (but female) we'd probably not got along to well, if she was kinda like me but different in crucial ways (as a parallel me would probably be if DC comics have taught me anything), I could see it working.
OT: I think it would be a case of picking a world to be on (does their world lose them, or does your world lose you...) and it would be a fairly normal relationship (just a normal hetero/homo thing depending on your preference and them being your ideal partner) after all, they are your parallel counterpart, you in a different universe, so even though they are you, they're not you, and even though they might be very similar, they're still not really you. *I don't know, my brain logic stuff is getting all fuzzy because of the glorious snow outside of my window*