You can think of i as representing a rotation of 90 degrees in the complex plane.

One thing I have problems visualizing is how quaternions define a rotation in 3d space. I get Euler angles (yaw, pitch, roll) and I can see how a 3x3 matrix can define a rotation/scaling transformation (it's just 3 vectors defining 3 axes defining a new co-ordinate system to map a point onto) but quaternions just don't make intuitive sense. I get how there have to be 4 components, because you can define any arbitrary rotation using an axis of rotation (a 3 component vector) and a rotation angle around it (scalar), and I know how to make a quaternion by plugging those things into a formula, but I just don't understand what results. How is the rotation somehow smeared across the other 3 components, to end up with 4 components that are each some weird hybrid of part axis, part rotation? Does anyone know an intuitive way to visualize this?