From A Topographical Dictionary of England by Samuel Lewis, 1848 (pp. 235-238)
FISHBOURN, NEW (St. Peter and St. Mary), a parish, in the union of West Hampnett, hundred of Box and Stockbridge, rape of Chichester, W. division of Sussex, 1¼ mile (W.) from Chichester; containing 295 inhabitants. It is bounded on the west by Chichester harbour. The living is a rectory, valued in the king’s books at £5. 10., and in the patronage of the Crown : the tithes have been commuted for £176. 16. ; the glebe comprises 14 acres, and the Dean and Chapter of Chichester receive a sum of £10. 14. per annum. The church is in the early English style. The remains of a Roman bath with a tessellated pavement were discovered in 1812, near the site of the Roman road here.
FISHBOURN, OLD, a hamlet, in the parish and hundred of Bosham, union of Westbourne, rape of Chichester, W. division of Sussex, 2 miles (W.) from Chichester ; containing 90 inhabitants.
A Topographical Dictionary of England, ed. Samuel Lewis (London, 1848), British History Online http://www.british-history.ac.uk/topographical-dict/england .
This publication was published between 1831 and 1859 in many editions. I can see no noticeable changes between editions for the entry for Fishbourne