One "mathematician's answer" is that that is what is required to make the quaternions a "normed division algebra."

That is to say, they follow most of the nice algebraic rules that real numbers do, except they are missing a couple features to fit the extra dimensions in. And it has been proven that the only such algebras are the one dimensional real numbers, the two dimensional complex numbers, the four dimensional quaternions, and the eight dimensional octonions. At each step, one has to give up another property that the real numbers have.

Also, the discoverer of the quaternions was William Rowan Hamilton. I don't see him mentioned directly by name above, so I'll put that here.