How can one intuitively think about quaternions? Quaternions came up while I was interning not too long ago and it seemed like no one really know how they worked. While eventually certain people were tracked down and were able to help with the issue, it piqued my interest in quaternions.
After reading many articles and a couple books on them, I began to know the formulas associated with them, but still have no clue how they work (why they allow rotations in 3D space to be specific). I back-tracked a little bit and looked at normal complex numbers with just one imaginary component and asked myself if I even understood how they allow rotations in 2D space. After a couple awesome moments of understanding, I understood it for imaginary numbers, but I'm still having trouble extending the thoughts to quaternions.
How can someone intuitively think about quaternions and how they allow for rotations in 3D space?

Thinking about quaternions as 4D is misleading. Quaternions are the union of a scalar and a 3-vector. Think: time and space. Space is a 3-vector. You can point in directions in space. Time is a scalar. There is a past (negative time) and a future (positive time) and now (0), but no ability to point in the direction of time.
Think of a blinking light on a train, each event having a time and location. These events can be written as quaternions. If the train travels at a constant velocity, you are seeing the addition of quaternions.
One can make movies out of quaternions. Examples are available on my web site, http://visualphysics.org
The rotations in 3D space calculations ignore time.
