The way to check and see if you're correct is to remove the other noun and see if it still sounds right. If you had said "Alex and me are going to the store" and you removed the Alex and part, would you say "Me is going to the store"?
… from the very start of The Alex Project I had determined that my professional approach would be rigorous in training and in testing my Grey. I had come from the so-called hard sciences, after all. I needed my data to be unimpeachable, to meet high standards of credibility. I wouldn’t let emotion cloud my judgment. I wouldn’t get too attached. My experience at the Clever Hans Battle made me even more determined to maintain as much of an emotional barrier as was feasible between Alex and me in order to keep that credibility intact, no matter how hard it would be. And it was hard.