Changing fields after PhD? I have two fields that I am interested in. Field A is historical and requires me to learn two new languages and I would love to focus on this first. Field B is more accessible, that is, there are no extra languages to learn or mathematical methods to comprehend. But it is a more contemporary discipline. I would like to move into this field afterwards.

How does one do this? The fields do not overlap much. 

People are often saying 'the value of a PhD is to show that you can do research, so once you have shown this, there is no need for a second PhD. You can move into a new area on your own now.'

However, I feel like it is incredibly important who one's supervisor was, and I am not sure how seriously people would take me in Field B. I would be 'not really one of theirs', as it were. I am also not sure how to get a job in Field B, after having done a PhD in Field A. I could apply for jobs in Field A and try to somehow move into Field B. But then, if a university hires me, to do research &amp; teach in Field A, which I studied for my PhD, how do I explain to them that I happen to have a paper ready in a completely different field? And how would I even find the time to write such a paper, considering that it will be expected of me to do research &amp; publish in Field A - because, this is after all what they hired me for.

Questions, Questions, Questions. :)
The most typical path for shifting your field after a Ph.D. is by means of one or more postdocs.  A postdoc is an opportunity for more training and is often used by people to move between fields.  

The challenge is finding the right postdoc (or chain of postdocs) to make the shift, which may be easy or difficult depending on the particulars of your fields. For example, shifting from computer science to systems biology is generally pretty easy; moving from Sanskrit literature to aerospace engineering would probably be quite difficult.

Once you have a more more permanent position, on the other hand, you are likely to find much of your time is consumed by the other responsibilities of that position.  For example,  if hired as faculty you will find much of your time is spent not as a researcher but as a teacher, advisor, department committee member, fundraiser, and administrator.  Combine that with something like a ticking tenure clock, and it can be quite difficult to change fields once you have passed the postdoc phase.