Perspectives on Riemann Surfaces

It's simply not true that Forster's approach to the subject is "dead".

If you are interested in algebraic geometry, then you'll be mostly interested in compact Riemann surfaces (which are basically the same as smooth projective algebraic curves over $\mathbb{C}$). The main theorems about these that a first course should cover are Riemann-Roch and Abel's theorem. Here Forster's treatment (eg stating Riemann-Roch in terms of sheaf cohomology and deriving it from Serre duality) is the standard modern treatment. It might not be my first choice of textbook (the subject is blessed with many good books), but it certainly would prepare you for algebraic geometry better than a course that is focused in analytic topics.

I have not looked at the other book you mention, but I would guess that it focuses more on open Riemann surfaces. While the third part of Forster's book covers these, it would not surprise me if the analysis people consider his treatment dated.

EDIT : In reply to your edit, I'm not exactly an algebraic geometer, but I'm a heavy user of Grothendieck-style algebraic algebraic geometry. Certainly there are people using hard analytic tools to prove things in algebraic geometry (eg the Siu school), but my feeling is that most people in the subject do not use them. Given the choices you have, you would probably profit more from a course using Forster's book.

I would recommend you to read Curtis Mcmullen's notes and course outline. I found them to be very useful even after I finished my Riemann Surfaces class in sophomore year(during which I used Miranda's book very often). For Riemann Surfaces with analysis, dynamics, etc you can of course again consult his lecture or Donaldson's book. I would also recommend the book "Lecutres on Riemann Surfaces" by Gunning, in case you are willing to read old books and have the patience to understand it carefully. The classical reference for researchers should be this.

Generally speaking even randomly picking a book and read is not so bad. I do not understand why you necessarily have to worry about one book is better or worse than the other because of "algebraic" "analytic" - you will find a synthesis! Every time you learn something by reading other's treatment of a subject you do not understand well enough. Even after I finished the Mcmullen's notes, I still find reading Lang's book to be very helpful. Also, Riemann Surfaces is a huge subject, with on going research in multiple fronts in different levels. Hoping to achieve a deep understanding of it in one semester might be difficult - because you have to do a lot of problems and usually they are not easy...If Lang resurrected and ask me a random question I am sure I will not be able to answer it...

Here is my 2-cents answer, I'm not an algebraic geometry expert, nor a Riemann surfaces connaisseur, but rather an end-user (singularities, algorithms), nonetheless I often regret not to know more about complex geometry, so here are some reasons why I think that analytic methods in algebraic geometry should not be underestimated, without even talking about strategy for your future.

Complex geometry and algebraic geometry are sisters — or brothers, I don't know, in French geometry is feminine —, and complex geometry is the older one. I think you cannot pretend being an algebraic geometer without knowing a bit of complex geometry :

• algebraic geometry constantly imports complex methods, and sometimes you cannot really understand them without understanding the complex part ;
• a big and interesting part of the algebraic geometry literature, old and modern, uses the complex language : Dolbeault cohomology, homology, complex residues, etc ;
• even Grothendieck uses complex geometry ;)

In the end you will know both sides, so I think the real question is in what order will you learn them ? Geometry is hard, you should choose the course you think you will understand better. Modern approach to geometry (analytic or algebraic) all have a serious pedagogical drawback : they kind of hide the geometry, and you can't always rely on the teacher to make it apparent, there is a big gap between the definition understanding, and the deep understanding. Choosing the right order can ease to fill the gap.

“students who have sat through courses on differential and algebraic geometry (read by respected mathematicians) turned out to be acquainted neither with the Riemann surface of an elliptic curve $y^2 = x^3 + ax + b$ nor, in fact, with the topological classification of surfaces (not even mentioning elliptic integrals of first kind and the group property of an elliptic curve, that is, the Euler-Abel addition theorem). They were only taught Hodge structures and Jacobi varieties!” — V. I. Arnold