It would be a great help to your understanding of the multiverse if you understand at least a little of Deutsch’s theory of knowledge. Therefore, chapter 1 of The Fabric of Reality would be a good addition to the original list for those who do not want to go too deeply into it.
The remainder of chapter 9 makes what to Deutsch is a central point, that quantum computing is another proof that the multiverse is real, but I found it to be quite difficult to understand. It requires the grounding in computing and virtual reality Deutsch gives in chapters 5 and 6 (“Virtual Reality” and “Universality and the Limits of Computation”), but even after this I don’t think he is explaining himself as clearly as he could here. And overall I found Deutsch’s writing on computability and Turing a fair bit less interesting than the parts on the multiverse and knowledge (though it obviously ties in with some of his points about knowledge), and so I do not include it in my recommended reading, even though in the end they are central to his overall picture, and I admit that the parts that I have grasped have been essential to my understanding. (Turing is another one of those concepts I’ve read dozens of times, but for one reason or another the significance isn’t entirely clicking with me.)
I suspect that many popular science readers will want to read chapter 12 on time travel whether I tell them to or not, and it is indeed fascinating. It relies to a great extent on what Deutsch previously explained about computation and virtual reality though (in chapters 5 and 6 and the rest of 9), which as I said I found not nearly as interesting and also difficult to follow in detail as Deutsch presents it. One can grasp his broad points though. He also presented a version of this idea in a March 1994 article for Scientific American called “The Quantum Physics of Time Travel” (co-authored with Michael Lockwood).