I was just curious would it be possible to have for example, a donought magnet be made to rotate (in a circle motion 360°) and simultaneously on its other axis to move back and forth?

 

You can certainly make it mechanically rotate,
with no apparent change in Magnetic-Field behavior,
but You can't make it rotate by means of an outside Magnetic influence
unless it has been Magnetized in a different orientation other than strictly axial,
in which case, the Magnetic-Field wouldn't exactly follow your drawn-lines.

Maybe by means of Eddy-Current-influence ????,
but it would probably be a very inefficient arrangement.
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Not directly IMHO, as you would need two different magnetic axes.

If you had 2 toroids attached, it could work.