What was the motivation for the complex plane?

I've read a bit about the history of the complex numbers, and many seem to credit Caspar Wessel with the idea of associating the complex numbers as points on a 2-dimensional plane (or at least the first to explicitly publish the idea). But what was the motivation behind this? Why did people make this leap, and what were their justifications for this?

edit: just to be clear, my question is only about the justification of identifying complex numbers as points on a plane, NOT about justifying complex numbers themselves.

Solution 1:

Complex numbers first arose in the works of Cardano and Bombelli in the 16th C who showed that formal calculation with square roots of negative numbers with the assumptions that:

(1) $(a+ib)+(c+id)=(a+c)+i(b+d)$

(2) $(a+ib)(c+id)=ac+i(bc+ad)+i^2bd=(ac-bd)+i(bc+ad)$

where $i$ is a formal symbol such that we may replace $\sqrt {-1}$ by $i$ led to the solution of many geometric problems. An example was the value of $x$ at the intersection of the graphs of $y=x^3$ and $y=15x+4$. This however led to uncomfortable questions like what kind of an animal is an entity $a+ib$? Merely defining it as a formal expression did not seem sufficient and it was a cause of great suspicion whether manipulation with such formal symbols was mathematically justified.

Wessel gave a formal mathematical meaning to the complex numbers by defining it as an ordered pair. His work was followed up by Argand and Gauss. This once and for all dispelled all doubts regarding calculations with complex numbers in the computations of solutions of cubic equations and other related problems.

Further edit: Wessel's motivation may not have been directly to give a meaning to complex numbers. He had a background in cartography and surveying and was naturally motivated to problems involving direction. He gave geometric meaning to "algebraic operations" on directed line segments, in other words he defined addition and multiplication geometrically for such directed line segments. Addition of course was done by the familiar parallelogram law, while for multiplication he defined an abstract unit $1$ and gave the now familiar similar triangle based definition. This also led him to define another directed line segment by rotating $1$ by $\pi/2$ and he observed that under his multiplication scheme this became $\sqrt{-1}$. Next Wessel established algebraic properties of these directed line segments, such as associativity, commutativity etc. Since these properties were exactly the same as those of complex numbers it was no great leap to make and conclude that the directed line segments were precisely one way to define complex numbers. As far as I am aware Wessel did not explicitly claim this though. However it is implicit in his work and Wessel was probably aware of it. Later mathematicians refined this idea and since an ordered pair such as in Cartesian or polar coordinated characterizes a directed line segment so the idea that a complex number is an ordered pair was born.