To answer your question. This book is even more complete and a bit improved over the first book. I don't believe there's anything in Machine Learning that isn't well covered, or correctly omitted from Probabilistic Machine Learning. This also has the benefit of a few more years of rethinking these topics. So between the existing Murphy books, Probabilistic Machine Learning: an Introduction is probably the one you should have.
Why this over Bishop (which I'm not sure is the case)? While on the surface they are very similar (very mathematical overviews of ML from a very probability focused perspective) they function as very different books. Murphy is much more of a reference to contemporary ML. If you want to understand how most leading researchers think about and understand ML, and want a reference covering the mathematical underpinnings this is a book you really need for a reference.
Bishop is a much more opinionated book in that Bishop isn't just listing out all possible ways of thinking about a problem, but really building out a specific view of how probability relates to machine learning. If I'm going to sit down and read a book, it's going to be Bishop because he has a much stronger voice as an author and thinker. However Bishop's book is now more than 10 years old an misses out on nearly all of the major progress we've seen in deep learning. That's a lot to be missing and it won't be rectified in Bishop's perpetual WIP book [0.]
A better comparison is not Murphy to Murphy or Murphy to Bishop, but Murphy to Hastie et al. The Elements of Statistical Learning for many years was the standard reference for advanced ML stuff, especially during the brief time when GBDT and Random Forests where the hot thing (which they still are to an extent in some communities). I really enjoy EoSL but it does have a very "Stanford Statistics" (which I feel is even more aggressively Frequentist than your average Frequentist) feel to the intuitions. Murphy is really the contemporary computer science/Bayesian understanding of ML that has dominated the top research teams for the last few years. It feels much more modern and should be the replacement reference text for most people.