As a post-doc you should have your own independent research project. However it is usual for it to have some kind of thematic connection to other work being done in the lab. Moreover, you should be using the time to develop your own independent ideas that you will take with you elsewhere (usually, to a faculty position of your own somewhere).
When I did a post-doc, I had a series of research tasks directly associated with my supervisor's research program. On the side I did some things on my own that really had nothing to do with her research at all, although they were in the same general area (conducting polymers). I managed to turn these into a paper or two 'of my own' - although as the PI her name went on the papers as well.
It is important to negotiate early with your post-doc advisor about what ideas can be taken with you when you leave. This way there are no misunderstandings later on.