Quaternions are the next level of hypercomplex numbers after the complex numbers. Complex numbers have the form ax+bi and Quaternions have the form ax+bi+cj+dk. They were discovered by Hamilton. Immediately after that Graves discovered the Octonions which have seven imaginaries and one real, which is the next hypercomplex number beyond the Quaternions. Hypercomplex numbers are generated by the Caley-Dickson process, which also generates a lot of defective algebras as well. The first three Hypercomplex algebras have special properties that the infinite number of Hypercomplex algebras starting at the Sedenion (fifteen imaginary numbers and one real) and going beyond that in an unfolding progressive bisection do not have. But each successive hypercomplex algebra loses an algebraic property as the hypercomplex algebras devolve. So Quaternions lose the commutative property, Octonions lose the associative property, and Sedenions lose the division property. After that there are no other important properties to lose. Complex and Real algebras have the same properties yet the Reals do not have conjugates.
Given this context, the Quaternions are the group structure of the fourth dimension. In the fourth dimension all knots untie. So there are not the same singularities in the fourth dimension as occur in the third dimension. And so this means that we can use the quaternions to do calculations that would normally generate singularities, because they have no singularities. This is useful for calculating the movement of robot arms. But unfortunately all the promise that Hamilton thought the Quaternions would have for mathematics did not pan out, and instead vector and matrix math was developed independently of Quaternions because they seemed to lack real physical use. Maxwell’s equations were first done in Quaternions, but then were translated out of that form, because there was not a clear use for the added complexity of the Quaternion computations. So Quaternions have turned into a kind of mathematical oddity, which does not have the power and usefulness of the complex numbers for physics.
However, physicists are getting interested in them again as they seem to have some promise of understanding the structure of the universe. Seehttp://7stone.com.
Also mathematicians are finding that they are connected to all kinds of other mathematical phenomena like the http://en.wikipedia.org/wiki/Fre….
Essentially the Quaternions are connected intimately with the dynamics of four dimensional space so where ever that becomes important then Quaternions are important.
Note: The Hypercomplex algebras are the main mathematical analogy for Special Systems Theory. See http://works.bepress.com/kent_pa… for a description of Reflexive Autopoietic Dissipative Special Systems Theory.