In 1763, having completed secondary school, he went to Denmark for further studies (Norway having at the time no university). 1778 he acquired the degree of candidatus juris. From 1794, however, he was employed as a surveyor (from 1798 as Royal inspector of Surveying).
It was the mathematical aspect of surveying that led him to exploring the geometrical significance of complex numbers. His fundamental paper, Om directionens analytiske betegning, was published in 1799 by the Royal Danish Academy of Sciences and Letters. Since it was in Danish, it passed almost unnoticed, and the same results were later independently found by Argand and Gauss.
Wessel's priority to the idea of a complex number as a point in the complex plane is today universally recognised. His paper was re-issued in French translation in 1899, and in English in 1999 as On the analytic representation of direction (ed. J. Lützen et al.).
Caspar Wessel's elder brother, Johan Herman Wessel was a major name in Norwegian and Danish literature.