Alexander Cross was born in Ireland c. 1806. He was appointed Surgeon to H.M.S. Columbine in the East Indies on 1 June 1838. H.M.S. Columbine was a sloop, built at Portsmouth in 1826 and of 492 tons. 71 officers and men, 24 boys and 20 marines were on board. John J. Crawford M.D. was employed as Assistant Surgeon. 
Alexander Cross was on the List of Surgeons of the Royal Navy fit for service in 1841.
Alexander Cross (1838) was appointed Surgeon Superintendent on the Equestrian to Van Diemen's Land in 1852. 
The following correspondence to the Royal Society Van Diemen's Land in 1854 reveals his residence as Rochester.
On a Specimen of Shark in the Museum of the Royal Society, Van Diemen's Land. By A. Cross, Esq., R.N. When at Hobart Town in December 1852, in the 'Equestrian,' convict ship, I noticed some fine specimens of sharks in the Museum there: one in particular, peculiar to the coast of Australia and New Zealand, which I identified as belonging to the genus Lamna, and which I marked Lamna cornubica; giving the specimen the same specific appellation as our English Porbeagle shark. There is the spine, tail, and jaws of the fish in the Museum; the last remarkable for its armature of long, thick, nail-like teeth, and the pointed form of the snout; hence the shark is called in Miiller and Keule's classification, Oxyrhina gomphodon, - it is the Tilueron of the Spaniards, a species of it being found in the Mediterranean, but not of the formidable dimensions of the Australian variety. In the British Museum they have the jaws of the latter, but no complete specimen. I write this to send to Hobart Town,
for the purpose of having the name which I placed on the specimen altered to Oxyrhina gomphodon, as I should be sorry to mislead our friends in the Southern Hemisphere by the term which I formerly applied to it, so I trust Dr. Milligan will correct the labelling accordingly, etc. Alexander Cross, Surgeon, R. N. Rochester, 23rd January, 1854. 1861
In the 1861 Census Alexander is aged 55 and resides with his wife Mary Anne age 40 and their four children all born in Rochester; and an unmarried sister. His occupation is recorded as Staff Surgeon R.N. and his birth place Ireland.
Alexander Cross was on the List of Deputy Inspectors General of Hospitals and Fleets, Retired (2 March 1864)
He is listed in the Medical Register of 1865 - Residence H.M. Dockyard, Sheerness, Kent. Qualifications Surgeon Royal Navy 1838.