The standard text is "Complex analysis" by Ahlfors, but imo there are better text out there.
I can recommend "Functions of one complex variable" by Conway.
If you want a book that is quicker-paced (and hence covers more topics), go for "Complex analysis in one variable" by Narasimhan.
First, you might want to look here for a collection of free books and notes on Complex Analysis. You didn't say what level (undergrad or graduate) you're looking for, so here are two suggestions, one for each level:
Undergraduate - I suggest https://www.amazon.com/dp/0071263284/?tag=pfamazon01-20 by Brown and Churchill. I studied (on my own) the first ten chapters (of the 7th edition) and really liked it. It's nice for self-study because it's almost free from typos and other errors. Also, most of the exercises come with solutions. However, most of the exercises are calculations, similar to calculus. Some of these exercises are still quite challenging. The text contains proofs, but few of the problems require proofs.
Graduate - I suggest https://www.amazon.com/dp/0387985921/?tag=pfamazon01-20. However, Lang's style annoys some. I think this is one of Lang's better books. I chose it because I'm interested in studying analytic number theory. Lang's book covers material relevant to that subject.
Ahlfors is a classic, and I think it's comparable to Rudin in exposition, though perhaps less rigid and somewhat more conversational in tone. It's fairly self-contained, though obviously familiarity with epsilon-delta arguments and knowledge of basic theorems from multivariable calculus will be helpful.
Other texts you may consider:
Complex Analysis by Serge Lang - This is the text I'm using for my current complex analysis class. I am personally not a huge fan because it starts off fairly slow if you've already had some experience with basic real analysis before. Also Lang apparently adheres very closely to the view that everything in complex analysis is described by power series - which is certainly true to some extent. The result is that by page 90, you'll only have seen one actual result of complex analysis, the local maximum modulus principle, via some rather ugly power series results. I think it's important to get across (and most likely review) the basics of power series, but I feel it's unnecessary to devote 40-50 pages to the topic. However, the text is self-contained, and Lang is a clear expositor.
Complex Analysis by Stein & Shakarchi - This is a superb introduction. Basic complex analysis is basically laid out in about 100 pages as the authors strive to introduce the subject as a simple collection of beautiful results. The exposition is terse and lucid, and the examples chosen are nontrivial. The authors tell you exactly what the crucial ideas are in complex analysis, and the approach as a whole is structured. To get the most out of this textbook, you will need to be very comfortable with basic analysis arguments.
I haven't seriously read Conway, but I know it's comprehensive and more modern in outlook in comparison to say Ahlfor's text.
A note on the problems. Ahlfors has a lot of good but not overly challenging problems (this can change drastically once you get to the later chapters, but I haven't gotten there). They'll definitely prepare you for problems at the level of say, Berkeley's graduate prelims. Lang's problems are actually pretty bad, as most of them are straightforward extensions (more like replications) of the examples in the text. Stein & Shakarchi has problems of varying difficulty, though most of them fall on the challenging side. The upside is that hints are provided for harder problems.
Rudin's book would be tough going for self-study. The exercises are very difficult, and you'd need to have a firm understanding of undergraduate real analysis. Also, if your primary interest is complex analysis, Rudin doesn't get to it until around p. 200.
"Visual Complex Analysis" by Needham is a truly beautiful book. However, it is so unconventional in its approach that it is not a good substitute for one of the standard textbooks if you are trying to match the material in a standard course. Great for supplemental reading in any case.