TITLE: What's the intuition for extending $\mathbb{C}$ to $\mathbb{H}$?
QUESTION [7 upvotes]: It seems to me that there is a clear, intuitive reason for extending the real number system to the complex number system. Namely, some polynomial equations that have no solutions in $\mathbb{R}$ become soluble in $\mathbb{C}$. When we do this, we lose almost none of the nice algebraic properties of the reals, and pick up some nice new ones along the way (e.g. fundamental theorem of algebra).
However, I cannot see such an intuitive reason for similarly extending $\mathbb{C}$ to $\mathbb{H}$, other than "because we can." In so doing, we actually lose the important property of commutative multiplication. So precisely what problems do we solve by moving from $\mathbb{C}$ to $\mathbb{H}$?
I am of course aware of the quaternions' numerous applications to 3D geometry; what I am really interested in here are the analytic properties they provide. Does quaternionic analysis offer results comparable to those of complex analysis?

REPLY [2 votes]: Sammy Black has offered an answer using clifford algebra. I will as well because I think he's missed the big point: that complex numbers and quaternions both arise from considering rotation operations in a real clifford algebra.
Consider the real clifford algebra $\mathbb G^2$, formed on top of $\mathbb R^2$.  The basis elements of the algebra are $1, e_1, e_2, e_1 e_2$.  Using the geometric product, the basis element $e_1 e_2$ squares to $-1$:
$$e_1 e_2 e_1 e_2 = -e_1 e_2 e_2 e_1 = -1$$
It's natural to consider $e_1 e_2$ as a geometrically significant quantity that behaves similarly to the imaginary unit in complex numbers.  The subalgebra of linear combinations of $1$ and $e_1 e_2$ is identical to the algebra of complex numbers.
But most critically, such objects arise naturally in rotation operations.  A rotation can be derived from a composition of reflections.  A reflection can be written in clifford algebra as follows:  a reflection across a line with normal vector $n$ maps a vector $a \mapsto -nan$.  So two reflections can be written as $(mn)a(nm)$ for another vector $m$.  The product $mn$ is also a linear combination of $1, e_1 e_2$, so $mn$ is "complex number".

The rotation argument works the same way for 3d, for the clifford algebra $\mathbb G^3$ that is built on top of $\mathbb R^3$.  In particular, the geometric product of two vectors $mn$ is in general a linear combination of four basis elements: $1, e_1 e_2, e_2 e_3, e_3 e_1$.  As before, each of these last three basis elements squares to $-1$, so they all correspond to imaginary units that are used in quaternions.