A persistent cough following an upper respiratory infection is quite common, but the great majority of such coughs resolve spontaneously within 12 weeks. The cough could be from sinus drainage, from infection precipitated asthma or from residual inflammation of your nose and/or sinuses from the "cold." If the cough persists for more than 12 weeks or if you develop any other respiratory symptoms, then you should see your doctor. Otherwise you would not need to see your doctor.