Alternate version of this question: is there any formula composed entirely of complex numbers that has a unique quaternion solution (with non-zero $j$ and $k$ components)?
I've struggled to understand quaternions for years now. I very much understand how to "get" from real to imaginary numbers, and then from imaginary to complex numbers. $\sqrt{-1} = i$ is an equation composed of only real numbers that results in an imaginary number. Then $1 + i$ is both an equation that shows how to "get" from real and imaginary numbers to complex numbers, and a representation of the final complex number itself. What I don't understand is the subsequent leap to quaternions.
The classic formula is of course $i^2 = j^2 = k^2 = ijk = -1$. But the existence of $j$ and $k$ do not seem "justified" by this in the same way that $\sqrt{-1} = i$ justifies $i$. The equation $i*i$ equals $j^2$, but it also equals $-1$. There is also $i = jk$, but this is really a way of getting back to complex numbers from quaternions, not the other way around.
I realize quaternions can be thought of as a 4-dimensional analogue to the 2-dimensional complex numbers, and that a complex number is just a quaternion of the form $a + bi + 0*j + 0*k$. But this does not help me understand them. It seems to me that the existence of the 1-dimensional reals and normal mathematical operators means that the 2-dimensional complex numbers have to exist, but there is no such equivalent logical connection between the complex numbers and 4-dimensional quaternions. Do you have to simply accept quaternions axiomatically?