Having a persistent cough after a chest infection is relatively common. Many people are worried about this residual cough and many seek medical advice very early after the infection is gone. For up to 6-8 weeks after the infection there may be a persistent cough and breathing may not be fully back to normal. This is expected, as the airways in the lungs are still recovering. If it is slowly improving, this is usually a good sign.
It is a good idea to rediscuss the matter with your doctor and seek medical advice if the cough persists over 2 months after the chest infection and does not show any signs of improvement (or if there is any worsening in your health status, as this may suggest a potential complication has occurred).