Doctor dies in the middle of the month (4)
I believe the answer is:
ides
'the middle of the month' is the definition.
(I've seen this in another clue)
'doctor dies' is the wordplay.
'doctor' indicates an anagram.
'dies' with letters rearranged gives 'IDES'.
'in' acts as a link.
(Other definitions for ides that I've seen before include "Fateful day for Caesar" , "Mid-month day in the Roman calendar" , "March 15; dies (anag.)" , "'''Beware the ... of March'' (Julius Caesar) (4)'" , "Mid-month date in ancient Rome" .)