Enquiring status of postdoc application
Is it a good idea to ask the person who one has applied to for a postdoc position about the status of that application


More of a necessity than a good idea.


  What should be the "reasonable" time period after the deadline of the application to enquire about the position. 


I'd say this depends on:


(As OP has suggested) The amount of time that has passed since the deadline. 
The amount of time before employment is set to begin.
The information published beforehand about the application procedure.
Whether it's the academics or the administrative staff whom you intend to ask
The number of expected applicants for the same position


but if you have not been instructed to await a reply in X time, it's perfectly reasonable to write and ask after, say, min(1 week + 1 week * round_up(log_8(number of applicants)), 0.25 * time between application submission and work start).

That's a very rough rule of thumb, and perhaps not super usable. Another way to put it is that you should write after having allowed a generous amount of time for your application to be processed, but no later than you need to make serious decisions based on the reply. It's not improper to say that you need to know because, say, you have to decide whether to move to another country or not, enroll your children in school for the next year etc.