To be clear: you normally would never try to visualize a number (if you feel the urge to do that, then that is probably a left-over urge from when you first learned about numbers by counting buttons or apples or cakes or sheep…which results in a really clumsy way of adding two integers (counting five apples, counting three apples, putting them together on one heap and counting them again) and getting ridiculous as soon as you get negative numbers…or fractions: divide one cake by four people, WTF! Then how do I divide two bananas by three carrots or one apple by seven sins? No wonder children have problems with learning to calculate…), say you have to calculate 5 * 3, you don’t imagine two points on a line, representing 3 and 5 respectively, now do you? And how would you visualize the multiplication?
The only reason we do that (visualizing numbers) with the complex numbers is to show that a multiplication in C describes a rotation and therefore a 2D rotation can be described as a multiplication of complex numbers. Once you understand that, it is a ‘small’ step to understand that a 3D rotation can be described by the multiplication of quaternions (again: quaternions are numbers!).
