I consider it very important that you become a expert person- gain tremendous mastery over one area in life. One single area, but, killer knowledge and expertise. It's really surprising sometimes when people who come from completely different background do miraculous work and/or become really important people in some amazingly different area. The guy who used to manage the whole DirectX graphics group in NVidia (where I worked for a year) was a PhD in nuclear physics. Yes. Or.. umm, a bright student here at IIT Delhi who is doing his PhD in algorithms had done his masters in VLSI! And these people have proved themselves to be much better than other people who may be there in the same field for quite a longer time.
We were discussing today what may be the reasons behind why this might have happened, and very important things came up. One was how the people who have done their masters/PhDs in mathematics or physics become potentially-good-at-any-analytical-field. They have become very strong minds, with really sharp abilities of critical thinking. They know how to attack problems, and give novel kick-ass out-of-the-box solutions. And our second conclusion was "yes, *this* is it"!!! This is exactly what a bunch of good profs, the work you do, your passion with your work, and the company you keep, end up doing to you. They do it to you and make it the person I mentioned above.
Isn't *this* something that you should be doing your PhD, and for that matter, even your masters for?
I started from a very different note than what I ended on...I hope it made some sense to you :)