Alexander Abercrombie was a son of the well-known Cape medical practitioner James Abercrombie senior. He went to Scotland to study medicine, qualified as a Licentiate of the Faculty of Physicians and Surgeons of Glasgow in 1860, and obtained the degree Doctor of Medicine (MD) at the University of Edinburgh the same year. His most important contribution to medical science appears to have been his Thesis on tuberculor leprosy, or the Elephantiasis Graecorum; with some general observations on that disease as it prevails at the Cape of Good Hope (Edinburgh, 1861, 32 p). Pages 21 to 32 were devoted to leprosy at the Cape.
Abercrombie returned to the Cape of Good Hope soon afterwards, was licensed to practice medicine there in July or August 1861, and settled in Wellington. He became a Fellow of the Royal College of Surgeons of Edinburgh in 1883 and served as a member of the Medical Board of the Cape of Good Hope in the eighteen-eighties. At this time he lived in Cape Town, and during the eighteen-nineties served as inspector of the psychiatric ward (then known as the "lunatic asylum") and leper hospital on Robben Island. He was a member of the South African Philosophical Society from 1879 to at least 1891.