Born at Leith on 17 March 1820, Dudgeon was a younger son of a timber merchant and shipowner there. After attending a private school he received his medical education at Edinburgh, partly in the university and partly in the extra-academical medical school.
Elected president of the International Homœopathic Congress which met in Atlantic City in 1904, Dudgeon did not attend because of bad health. He died at 22 Carlton Hill, London N.W., on 8 September 1904 and was cremated at Golder's Hill, his ashes being buried in Willesden cemetery.
A utopian science fiction novel, Colymbia (1873), was a response to Erewhon of the previous year: Samuel Butler was a patient and friend.