There is no prohibition of work during Chanukah because it is not a biblical holiday and the Sages never instituted an issur melacha on these days.
However, it is customary that in the first half hour after the Chanukah candles are lit, women refrain from certain chores, and there are different customs regarding which chores are included in this. Some say that this custom applies also to men.
Other than that, no restrictions – aside from the fact that some people (if they are able to) leave work early to allow for a timely candle lighting.