Quaternions. Problem with understanding essence of. I can't calm down when I can't realized of principles of working something. Quaternion is that place where I spend а few days!
Unfortunately, not one lectures or books what I saw or read don't explain principe of working of quaternion in 4d, as it understood sir Hamilton. Yes we works in 4d and it's so hard to explain, but if quaternions is a expand of complex numbers, it's must works simillary.
For example, why we use in Euler formula for quaternions sin for any three coordinates of vector, if they perpendicular to each other in imagine space? At the same time we use cos only for real part. 
$$
ql = \left[cos (\frac{1}{2} \theta) , sin (\frac{1}{2} \theta) (i+j+k)\right]
$$
I explain it for myself as three Cartesian plane with the one common real component which defines a vector position in 4d. This is my attempt to somehow imagine a quaternion. Is it far from reality? 
Why we disregard of real part of coordinate in first part of Euler formula?
$$
ql = \left[cos (\frac{1}{2} \theta) (w), sin (\frac{1}{2} \theta) (i+j+k)\right]
$$ 
Why we need multiply quaternions by involving sandwich if we don't need that in complex number multipliction?
$$
p^. = qpq^-1
$$ 
May be I'm on wrong way to understanding of quaternion by equating it to 2d complex number, let me know.

I found this helpful article for basic understanding of quaternions.
