Alexander Morton was born on September 11, 1854, New Orleans, Louisiana.

He was a naturalist and museum director. He moved to Australia as a child with his father Thomas William Morton, who migrated to Queensland as general manager of the Manchester Queensland Cotton Co. Alexander was a seaman for about two years. He visited England and Europe briefly but returned to Australia to study the natural sciences. He was a curator's assistant at the Australian Museum, Sydney (1877-1882) and took part in expeditions to New Guinea (1877), the Solomon Islands (1881), and Queensland and Lord Howe Island (1882). In 1884, he became Curator of the Royal Society of Tasmania's museum and its library in Hobart (renamed the Tasmanian Museum, Art Gallery and Botanical Gardens in 1885). By 1891, he had reorganized the museum, using the latest British Museum labelling methods, and evolved a highly regarded system of classification and arrangement. He was made Director of both the Tasmanian Museum and Botanical Gardens in 1904. He was also honorary secretary of the Royal Society from 1887 to 1907. He helped establish the Queen Victoria Museum and Art Gallery at Launceston and acted as an honorary curator in 1891-1896. As general secretary of the Australasian Association for the Advancement of Science (Hobart, 1892), he edited its reports and papers. He was a Fellow of the Linnean Society, London (1889).

In 1884, he married Caroline Eliza Mills (1856-1914). He died on May 27, 1907, in Sandy Bay, Tasmania, Australia.