I have worked through Ahfors for an introduction to Complex Analysis. I am seeking advice on the books to read to learn more Complex Analysis. At this point, one name suggested to me has been Conway. My background includes Analysis at the US undergrad level -- analysis in euclidean spaces, point set topology, measure theory, basic functional analysis and differentiable manifolds.
John Conway's two books on complex variables are very nice.
Some relatively standard "2nd course" books are (or at least at one time have been)
Conway, Functions of One Complex Variable II (1995)
Hille, Analytic Function Theory, Volume II (1962)
Saks/Zygmund, Analytic Functions (1952)
Veech, A Second Course in Complex Analysis (1967; 2008 Dover reprint)
I don't know much about Conway's book, but I do know that his earlier "Volume I" is a classic. I've had a couple of courses out of it, one taught by Conway himself. Hille's book is very nicely written with lots of historical and tangential digressions (his "Volume I" is perhaps even more distinguished in this way), and Veech's book was reprinted by Dover a few years ago so it's probably better known now than it was 10 years ago (i.e. you can probably find a few comments on the internet about Veech's book). Saks/Zygmund is a classic, probably too old for primary study now, but for what it's worth, this was the text used for the advanced graduate level complex analysis course at Univ. of North Carolina in the 1970s (offered every other year, with Ahlfors being the text for the beginning graduate complex analysis course that was offered every year).
Besides considering these books, you may want to begin thinking of what your "end game" interests are. For example, if you're a hard-core analysis person who doesn't like pictures, then you might want veer towards something like
Boas, Entire Functions (1954)
Holland, Introduction to the Theory of Entire Functions (1973)
On the other hand, if you really like the geometric aspects of complex analysis, then you might want to veer towards something like
Väisälä, Lectures on $n$-Dimensional Quasiconformal Mappings (1971)
Iwaniec/Martin, Geometric Function Theory and Non-Linear Analysis (2001)
Krantz, Geometric Function Theory (2006)
Complex analysis is a HUGE field, and at some point you'll have to start narrowing your focus. Any one of the following topics (picked out of the blue by someone who is not all that knowledgeable about the subject) is more than comprehensive enough for a lifetime of study and research: several complex variables, zeta function stuff (Dirichlet series, Riemann hypothesis, prime number theorem), summability theory, boundary properties of functions, distribution of zeros of entire functions, iteration of functions, Blaschke products, etc.
My favorite and my first choice for complex analysis is
A Course in Complex Analysis, From Basic Results to Advanced Topics Wolfgang Fischer, Ingo Lieb, 2012.
It is concise and contain enough things from basic to advanced. Another suggestion is
Complex analysis, Theodore W. Gamelin, 2001.
If you want a fast, strong book like Ahlfor's book, you can take a look at
Complex Analysis, Elias Stein, Rami Shakarchi, 2003
You know, Ahlfors himself wrote such a book.
Prerequisites for the book are exactly the topics in Ahlfors's Complex Analysis