For people trying to understand quaternions, the faster way to intuition is understanding axis-angle first. I think axis-angle is totally intuitive. First pick a direction (some 3d unit vector) and then rotate around that direction by some amount (so some scalar). Then convert those four numbers to quaternions
qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)
where (ax, ay, az) is the unit direction (ax^2 + ay^ + az^2 = 1) and angle is the amount you want to rotate.