What exactly does a quaternion represent? I am wondering a few things about quaternions.
Firstly, besides the $ijk=-1$, etc definition, what exactly is a quaternion? What is their geometric representation and how does this differ from the representation of complex numbers on the complex plane? Also, is the $i$ in $ijk$ the same $i$ as in imagary numbers? If not, what exactly is the difference?
 A: Quaternions are a way of representing vectors in 4-dimensional space: the vector corresponding to $a+bi+cj+dk$ is simply $(a,b,c,d).$. This is, naturally, not as easy to visualize as the situation with complex numbers. But you can still imagine things in terms of, say, amplitudes and phases; it's just that unit quaternions don't lie on the circle, as with complex numbers, but on the three-dimensional sphere. This may be an unfamiliar object, but it isn't so bad: in the same way as the two-dimensional sphere is just the plane with an extra point added outside, the three-sphere is just three-space with an extra point added, so that unit quaternions can essentially be visualized. It's important to note that this visualization doesn't correspond too well to the other one. A unit quaternion, so $a+bi+cj+dk$ where $a^2+b^2+c^2+d^2=1$, corresponds to a vector in three-space after applying a stereographic projection.
Unit quaternions can be identified with rotations of three-dimensional space, which is often the best way to think about them. Specifically, take a point in the three-dimensional sphere. If it's either the origin of three p-space or the extra point, it represents the trivial rotation. Otherwise it lies on a unique line through the origin, and represents the rotation of three-dimensional space about that line through four times the arctangent of its signed from the origin, in a direction determined by the orientation of three-space. There are then exactly two points representing every rotation, namely those on the same line representing clockwise and counterclockwise rotations through angles summing to $2\pi$. 
This representation by rotations is again actually an analogue of that for complex numbers: unit complex numbers are just points on the circle, which is the group of rotations of 2-dimensional space. So geometrically the quaternions reproduce many phenomena first seen in the complex numbers, with 2 extra dimensions added everywhere. You can interpret both $i$s as the same, if you want.
A: This is a slightly complicated question but put simply. C is the field over R such that it is of the form (a+bi) where (a,b) are in R. The quaternion is a further extension where numbers are of the form (a+bi+cj+dk) where (a,b,c,d) are over the Reals. It has the properties i^2 = -1, j^2 = -1 and k^2 = -1 and ijk = -1. From this it can be derived that IJ=K,JK=I,KI=J and JI=-k,KJ=-I,IK=-j. This gives the largest trivial difference between the quaternions and other fields namely that it is not commutative. This is why the quaternion are usually represented as matrices. I in the quaternions can be thought of as i*(2*2 identity matrix).
