TITLE: Between complex numbers and quaternions?
QUESTION [5 upvotes]: Complex numbers are $a+ b i $; Quaternions are $a + b i + c j + d k $. So, do there exist numbers like $a + b i + c j$? Here $a$, $b$, $c$, $d$ are all real.
Did Hamilton consider such a case?

REPLY [2 votes]: Compare $b$i+$c$j+$d$k with the vector $(b,c,d)$ and for multiplication of such entities compare quaternion multiplication with the vector product. You will see that there is a clear relationship between the two.
Hidden within the quaternions is this structure which does relate to three dimensions.
So though you can't do everything you want with three components, you can get somewhere.