Is there an exposition of complex analysis firmly separating the algebra, analysis, and topology? Complex analysis seems to work because of the interplay between algebraic geometry over $\mathbb{C}$, and analysis and topology exploiting the fact that $\mathbb{C}/\mathbb{R}$ happens to be a quadratic extension.  Is there an exposition of elementary complex analysis (maybe at the level of Ahlfors's Complex Analysis) that very carefully separates these things out, perhaps pointing out at each step what would be the same or different when working over other fields or field extensions?
 A: A book that makes a real effort to separate the analytic and topological aspects of complex analysis is Steven Krantz and Robert Greene's Function Theory of One Complex Variable. Like most of Krantz' other textbooks,it's extremely well written and takes the student deep into a graduate course in complex analysis.The prerequisites are minimal,too-just a good careful background in calculus of one and several variables.Indeed,one of the original touches of the book is the authors emphasize the connections between multivariable calculus in the plane and complex analysis in one variable.
I think you may find it exactly what you're looking for. Also, I'd like to second the recommendation of Jones and Singerman-it's a wonderful book focusing on a modern presentation of the relationship between geometry and complex analysis. Krantz/Greene generally focuses on the purely analytic aspects of the subject-as a result, the 2 books complement each other beautifully.I think that combination will work very well for a graduate course. 
A: Strictly speaking the answer is "no", there is no such book. But this is perhaps a good approximation to what you are asking:
MR0890746 Jones, Gareth A.; Singerman, David Complex functions. An algebraic and geometric viewpoint. Cambridge University Press, Cambridge, 1987. 
