His first major work was the improvement of the medieval bridge at Rochester. The bridge was widened and the two central arches merged into one to provide a wider channel for shipping. The work was not completed until 1824, by which time Alexander had been dismissed from his post as engineer to the bridge. He was the principal architect of Dartmoor Prison and Maidstone Prison, two of the oldest gaols still in use in the United Kingdom.
In 1799 he carried out a detailed survey of Rochester Cathedral, and recommended a programme of repairs, which was begun in 1801.
Alexander was the surveyor to the London Dock Company between 1796 and 1831 and was responsible all the buildings at the London Docks during that time, including the Pennington Street Warehouses and the London Dock House (the dock basins themselves were by the company's engineers, including John Rennie).
In his capacity as surveyor to the Trinity House he built a number of lighthouses, including the High Lighthouse at Harwich (1818), and others at Holyhead, Farne Island, and Lundy Island, the latter built on older foundations in 1819.
Other works include Mote House near Maidstone, built in the 1790s for the 1st Earl of Romney; Coley House near Reading and extensions to Inigo Jones' Queen's House (then the Royal Naval Asylum) in Greenwich, London.