I am confused with what Hamilton's equation means. I understand that the coordinates span 4 coordinates but I don't see how the quaternion product formula works. For example, if we break apart the quaternion product formula into simply i^2 = ijk, this intuitively does not make sense. Is there a way to visualize this?
I found this video
Also what does the real part of the quaternion represent? Is it always zero?
An interesting topic I found was about how increasing the dimensions of these complex number systems results in losing some algebraic property.
Real -> Complex : Loss of ordering
Complex -> Quaternions : Loss of commutativity
Quaternions -> Octonions : Loss of associativity
More in depth explanation can be found on this Wikipedia page.