I'm in the last year of my undergraduate studies. I recently decided to become a mathematician and I will apply for a two year master degree as a preparation for a PhD. Looking to various master programs I noticed that some of them tend to oblige the student to complete some basic work and then to specialize in one area, taking courses and seminars in order to gain enough experience in the field for a final thesis on some actual research topic.
I understand that this is a good way to start some research work and that an advanced thesis could be a good presentation for a PhD program, but I'm not sure if it's a good choice on the long run. For what I know a mathematician changes field during his career and I believe that taking many courses in different areas, building a basic understanding of a wide range of branches of mathematics, could be a better choice for this early stage, as a strong basis for a long and rich mathematical life. After this I will be able to appreciate very different mathematics and to do a thoughtful choice of an area of expertise to work in during PhD.
I'd like to compare my beliefs with those of some experienced mathematician. What would you care about in the beginning of your mathematical career?
P.S. when I talk about a wide range of mathematics, I'm particularly inspired by the possibility to cover with courses and individual work basic material from most of the branches described in the Princeton Companion to Mathematics, here is an index
NOTE I have a good undergraduate background. As a reference, by the end of this academic year I should have covered a representative part of the content of Rudin's Principles/Real and Complex/Functional Analysis, Lee's Topological/Differentiable/Riemannian Manifolds and Lang's Algebra