So I am assuming in the process of traveling together Me will give Clara the other chip so she can NOT die (though Clara seems to be doing fine without any help).  Did the Me at the end of the universe already have a “book” about her travels with Clara or is this new in her time stream?

Maybe “me” and “Me” are both right (in a sense).  Maybe Ashildr is his mother.  By the way, I loved the episode.  And I love playing the long game.