What is the best way to supplement a complex variables class to make it more complete for a math major? For the upcoming semester I plan on a taking a “complex variables” course that many people, including myself, would not consider a true complex analysis class. I know that the course will likely use a text similar those by Saff & Snider or Brown & Churchill because it is more of a survey class meant to give the basics for leading into to true complex analysis classes and giving the appropriate tools for physicists and engineers. As someone interested in theoretical mathematics, I naturally want to expand my knowledge beyond what is taught, see a more rigorous presentation of the material, have applications leaning more toward number theory than physics, and see topological constructions in action. I know that Ahlfors’ Complex Analysis is the a very common text instructors and students turn to for what I am looking for, but it is very expensive ($200 USD + for a ~300 page text), and I have heard people describe it as “difficult” for independent study unless you really know what you’re doing beforehand. Is there a better text for me to follow? I see that MIT has its 18.112 course (Functions of a Complex Variable), an undergraduate level course based on Ahlfors, listed on OCW, so I would have something to follow and test myself on, but I would prefer to not use Ahlfors. I have seen recommendations to other people to use Visual Complex Analysis for self-study, but this book is still more directed at undergraduate physics students and the like. 
What are the best alternatives to a text like Ahlfors? Which are the best suited for independent study for someone working alongside a less mathematically rigorous course? Which are the more comprehensive? Are there any that follow naturally from where books like that by Brown & Church leave off? Which are the most comprehensive, and are there any that lead into analytic number theory or give a taste of complex analysis in several variables? 
 A: I recommend Stein and Shakarchi's Complex Analysis. It's clear, easy to read, and gives a proof of the Prime Number Theorem. It also has a little more material on analytic number theory in the last chapter, on representations as sums of squares (via theta functions).
I would supplement this with the material on Cauchy's theorem in Ahlfors' text. Stein and Shakarchi give a handwavey proof of a simple version in their text, which I think is appropriate for a first read, and leave some remarks on more general versions to their appendix. I don't remember if they actually prove a more general version in that appendix, but Ahlfors definitely does. 
A: I can't recommend Visual Complex Analysis enough -- I would say it is one of the ten best mathematics textbooks ever written.   I'm not sure why you think this book is directed towards physics majors.  I read it as a graduate student in pure math, after having taken two graduate courses in complex analysis, and I felt like it provided me with significant insight into complex analysis that I hadn't gained from either course.
A: For an introductory course or independent study I recommend Ash & Novinger for math majors.  Unlike Brown & Churchill, for instance, it is not afraid of using aspects of real analysis (beyond open/closed sets, limit points, etc) such as compactness in its proofs.  Apparently, you will enjoy the absence of applications to problems in heat conduction.  The final chapter proves the Prime Number Theorem, so there's your number theory.  I doubt it will meet your topology demands, but for that you probably should be in a second course in complex analysis in the first place.
Ash & Novinger is also clearly written.  Perhaps a superficial problem with Ahlfors that people don't like to discuss is the old school typeset and endless strings of paragraphs with no intervening figures.  For this modern reader I find it fatiguing to read.  Like others, I also find Ahlfors opaque, and simple concepts are drenched in unnecessary and often unclear verbiage.  Contrast this with Rudin's Real Analysis.  For all my problems with that book's terseness, it is beautifully written and pleasant to look at.  Is it so wrong to want a math book to be aesthetically pleasing to the eye?  I like chummy exposition and proofs clearly delineated.
Lastly @jmracek:  Gamelin refers to Goursat's Theorem as aesthetically pleasing but useless, not the extremely useful Cauchy-Gorsat Theorem.
A: I have always really liked Gamelin as a reference on complex analysis.  All of the basics will be found in that book.  I think it is an entertaining read, as Gamelin's sense of humour often shows.  For example, he refers to the Cauchy-Goursat theorem as "aesthetically pleasing as it is useless".  In all seriousness, there are a lot of problems, together with hints and partial solutions in case you get stuck.  
One complaint I have heard about the book is that proofs are incorporated into the text, and not part of a separate proof latex environment.  Some people seem to think this makes it unclear when a proof is beginning or ending, but I personally have had no problem with this.
