Born in West Linton, Scotland, James Martin Alexander studied medicine at the University of Glasgow. After graduating in September 1941, he held house physician posts at the Royal Hospital for Sick Children, Glasgow and at EMS Hospital, Gleneagles, Perthshire until November 1942. After two months in general practice, he joined the Royal Naval Volunteer Reserve as a surgeon lieutenant and was on active service until at least the end of 1945. His permanent place of residence was Canterbury, Kent.
Whilst on naval service, Alexander spent several periods stationed in Sydney. He took the opportunity to attend postgraduate teaching sessions and at least one course in medicine, also attending ward rounds at St Vincent's and Royal Prince Alfred Hospitals. He sat for and passed the Membership examination in September 1945, the first Briton to do so.
Later, he held a house physician post at the Hospital for Sick Children, Great Ormond Street, London, gained the DCH in 1947 and married. He died in 1951 of tuberculosis, presumably a tragic legacy of his naval service in submarines.