As a background, I am a beginning graduate student whose background is primarily algebra and algebraic geometry. I have some background in analysis, but it is weaker. In particular I know the basics of measure theory and complex analysis but only at an undergraduate level.
I am really interested in the Riemann-Roch theorem, and am approaching it from an algebraic geometry perspective at the moment, but I thought the best way to learn complex analysis would be to approach from the Riemann surface point of view.
With that in mind, I am looking for a complex analysis/complex geometry book that is fairly self contained in terms of complex analytic background. Since I want to learn some complex analysis as well, I would like something that doesn't shy away from analytic arguments when necessary, but I would like something with a definite geometric flavour.
I feel like the standard answer here is Griffiths and Harris, so perhaps it would be good if people could give me an idea on how this book would serve me for this purpose? I got the impression is was much more algebraic geometry than complex analysis though. The other is Schlag's Complex Analysis and Riemann Surfaces if anyone is able to give me an idea of how that would serve.
Does such a book exist, or an I asking too much? To put it shortly, a self-contained complex analysis and complex geometry which doesn't shy away from complex analytic arguments but has a geometric slant with the view towards proving the Riemann-Roch theorem for complex manifolds.