I checked the Blitzer book I've taught from, and see that, indeed, it doesn't use any notation for conjugates, because it doesn't do any more with them than use them to simplify expressions or point out that solutions to quadratic equations come in conjugate pairs, neither of which needs a notation.

Clearly your new book is at a higher level, so you may need to go back to where they introduced complex numbers and read carefully, to see their notation and conventions.

In the solution you quoted, they don't take the time, as one would in a proof, to define variables or justify steps, trusting that you know what to expect: When they replace z with a+bi, they are implicitly saying, "suppose that the complex number z has components a and b", and so on. So you have to build up your expectations to match theirs, by reading slowly and carefully, even when you think the topic is one you already know. Probably there has been an example that demonstrated these ways of thinking.