So, you all know how with string theory (or M-theory) there are an infinite number of universes. And how in all those universes it is statistically possible there are multiple yous doing slightly different things. Now most SF people focus on this last point and start doing the the "character crosses over into alternate universe/reality and must 1) realized what happened and 2) adjust or get back to their home universe."
But what they miss is that with the infinite number of universes it is also statistically possible that if you could actually jump from one universe to another (or from one brain to another), chances are you'll end up in a completely unknown place that may not even play by the same rules of physics (this is sometimes used as the "hyperspace" to get FTL travel). Within an infinite number of universes, the universes with alternate Earths, and specifically you, in it is still a vanishingly small number (infinity is a really huge number, I mean, really, really huge).
For this experiment we'll ignore the also highly likely chance that you aren't anywhere near each other if you were able to build a bridge between those membranes.
However, if there are an infinite number of other yous out there, surely another one of them (at least) is/are also trying to get to you. How could you (as the mad scientist you are, chancing the possibly of paradox issues and creating massive-Earth-eating blackholes) work in collaboration with that other you who has also realized another you may be trying to reach them to build a membrane/universe bridge that will allow them to meet.
Or in other words, you're not building the bridge one way across the divide. You're goal is to meet that other you halfway.
Bonus points Okay, write it as first person or tight/limited third. However, switch between yous every break. Figure out how to signal to the reader which you they're following quickly and clearly. Then, when the "break through" discovery happens, switch characters mid chapter (alternating them in a few fashion). Finally, when they meet, they see each other as completely different (ie. not in the "one of us" manner most universe switching deals with). And while the reader knows they are the "identical twin", the characters don't realize that.