I was wondering if it's possible to use the same research question, just using different methodology and a more expanded research area.I don't mean if you can just extend the topic of your Master's thesis into something that you're going to research on, in your PhD (as in your MA thesis becoming one or two chapters of your PhD thesis).
Researching the same topic for a Ph.D. as a Masters is entirely possible, and not necessarily even all that unusual (it depends on how you define "same"). If the Masters is a first set of results on the topic, and the Ph.D. then pursues a significant advance thereupon, that would be an entirely reasonable progression of research.
Certainly, if this is fine with your advisor and committee and there are enough "blank spots" left in this particular area.
(Some researchers could be accused of not only devoting their Master's and Ph.D. theses, but their entire research careers, to a single research question. These researchers, however, do not tend to be too highly regarded. Some switching of focus in one's early career is beneficial.)
My MSc and PhD were of the same topic - even had the same supervisors!
The way we approached it was to have the PhD research follow on one of the more complex suggested avenues of further study and one of the more surprising outcomes that I presented as part of the MSc. This made writing the research proposal (required at my University) much easier to write, justify and ultimately, be approved.
Essentially, as @jakebeal states in his answer:
the Masters is a first set of results on the topic, and the Ph.D. then pursues a significant advance thereupon
My PhD was built out of a complex extension of one of the outcomes of the MSc
Along with other answers here, Im going to add another yes. I'm doing that right now - after taking a detour via a Doctoral Training Centre, I decided to go back to where I left off. Best decision I've ever made.
Firstly, this was good because my masters supervisor was simply brilliant; as a PhD supervisor he continues to be enthusiastic, push me in the right directions and give me the freedom to explore whatever area I think may come in handy later on.
Secondly, I really enjoyed my masters. It gave me a good amount of background knowledge in a relatively niche field. I've abstracted from my earlier work, and now I get to push forward and get some much deeper and more general understanding of the original work.
Thirdly, the skillset. My masters project was in condensed matter physics (theoretical), and simulations formed a big part of that. I had built up a codebase, and went through the process of figuring out how things patch together. When I came back to the project, the first thing I did was start from scratch, knowing the limitations of my earlier code, and made things more efficient and flexible. Whilst this applies to code, it applies in general - you learn tricks, you hardwire your brain to think in a certain way and you make many, many mistakes which you learn from.
Note, though, that a DIRECT continuation isnt always wise. I'd recommend spending your early PhD days reading, absorbing, and attacking new problems. Combine that with your knowledge from your original project and you will be unstoppable.
Many people will tell you that it is better to explore the more general area, chsnge supervisors and move around (maybe abroad). Sometimes, this is good advice - your later career may benefit from having worked closely with multiple people, it'd make for a more colourful CV and your skillset would be wider. On the other hand, your jump start would get you more publications earlier on, your supervisor can arrange/encourage collaborations.
One final note: above all, if you're happy with your project and happy in your research environment, don't take that for granted. It doesnt mean you should stay there forever, but it does mean you should choose your next place carefully.