Transitioning from a PhD to Postdoc. Is it acceptable to stay with the PhD advisor for 6 months to a year after graduating?
In mathematics and in Northern Europe I have seen more than a few people do this. A grace period after graduation is not rare. But it is expected that you spend time elsewhere, preferably abroad, either employed or on long and frequent visits, or that you are very good. (Even if you are very good, there are probably other very good people seeking employment, so travelling is still a good idea.)

More importantly, going elsewhere gives you perspective:


New ways of teaching.
New fields and problems to solve and methods for solving them.
New contacts and collaborators.
If going abroad, usually a new language and a more-or-less new culture, depending on how far you go.


In most life situations, it is easier to travel now then later, even if you intend to come back. So if your situation allows, consider seeking for positions elsewhere. Current co-operators of your current research group would be especially useful, since that would probably allow you to contribute to the ongoing projects in some way, too.

Also, unless you specifically aim for it, you will never not have unfinished projects going on, though this is probably field- and personality-specific.