I found a listing on Google books for a book containing the desired English translation, together with some biographical information on Wessel, and entitled On the Analytical Representation of Direction, but I cannot find any source that actually has a copy of this in stock. Needless to say, I can't find it anywhere else either.
I would very much like to see the language and ideas of the man who first conceptualized the complex plane, as well as whatever else he felt was relevant! Has anyone read this paper? Does anyone see any good reason why it should be so unavailable?