To understand quaternions better, I suggest you take a look at geometric algebra, see for example this video, this book, or Google around for better books/resources.
Basically, in 2D (resp 3D) a rotation by a quaterion is actually just a double reflection about two 2D lines (resp 3D planes), the rotation will be the double of the angle between the two lines (resp planes). You can easily get a feeling of that by drawing it on the piece of paper.
Furthermore, the lines (resp planes) can be rotated without changing the effect of the rotation. So if you have 4 lines (planes) for doing 2 consecutive rotations, you can always rotate the first pair so that the second line of the first pair coincides with the first line of the second pair (or vice versa = associativity). Now a reflection about two coincident lines (planes) is an identity operation, meaning that any sequence of rotations can be collapsed to a single rotation (in matrix terms, the product of matrices is a single matrix).
Likewise, a translation is a double reflection about two parallel lines (planes), the distance will be the double of the distance between the two 2D lines (3D planes). This is known as dual-quaternions. Again you can move the two lines without changing the translation, collapsing any sequence of translations into a single translation.
You can now combine these two, coming to the geometrical conclusion that any sequence of translations and rotations can collapse into a single translation and rotation, just by rotating and translating the line (plane) pairs.
But be warned, geometric algebra is still considered "alternative math", not well accepted by the mainstream community, just as reactive programming was when I discovered it... It is very addictive, because it merges so many different mathematical areas into a single beautiful framework. Actually, by just starting with a linear space and adding a single property (that a geometric product * exists for which v * v is a real number), the full and rich geometric algebra follows.