Is the existence of quaternions as an extension of complex numbers an axiom in a way that complex numbers as an extension of the reals aren't? 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?
 A: Edit This is more a lengthy comment than an actual answer. I realized it doesn’t tackle the actual question, but I don’t want to take it down, since I learned something from the comments it attracted, so maybe someone else does as well.
I think the complex numbers $\Bbb C$ and the quaternions $\Bbb H$ were discovered / are motivated out of very different reasons.
In the case of complex numbers the question was, what happens if one freely adds a solution $i$ to the real polynomial equation $x^2+1=0$. It turns out that this is not some abstract trick, but can be realized in form of the complex numbers. Now it is a theorem that not only every real polynomial decomposes into complex linear factors, but in fact every complex polynomial does as well. This is to say: the complex numbers are algebraically closed.
Now having the complex numbers at hand, you can play a bit with them and notice that complex multiplication describes rotation and scaling of the plane, which makes dealing with such transformations of the plane extremely easy. While developing the mathematics of mechanics it became an important question, whether a similar algebraic description of rotations in the 3dimensional space was possible as well. In this context, Hamilton discovered the quaternions $\Bbb H$. Note that this has nothing to do with polynomial equations anymore.
Now the final question becomes, whether higher dimensional transformations allow a similar algebraic description as those in dimensions two and three do. After some translations one can phrase this as the Hopf Invariant One problem, which was resolved by Adams using tools from algebraic topology: The only calculi of transformations are those given by $\Bbb R, \Bbb C, \Bbb H$ and $\Bbb O$, the last one being the Cayley-Octonions.
