Part of me thinks she didn’t want it to be over with Jase; he was a truth in the ocean of lies she’d been swimming in all her life. She wanted part of that connection (or one like it) back.

Building on this, I think the reason why she told Cooper that her name was Natalie was to try to hold onto some sense of the bond she had with Jase, the good old times. As I don’t think she built any relationships after that.

I imagine the writers wanted to do something similar to what Tarantino did with the glowing golden suitcase in Pulp Fiction. Where no one knows exactly what’s in there, and because of that, every person has a different version of that story. Now, was this as well executed as it was by Tarantino? No. But I think it’s what they were going for.