By the Cayley-Dickson construction quaternions can be written as $q=z+wj$ and octonions can be written as $x=q_1+q_2l$. Is there any generalization of the real part and the imaginary part such that $$\Re(q)=z,\ \Re(x)=q_1,\ \Im(q)=w,\ \Im(x)=q_2.$$
Though we can not distinguish imaginary units in general, in this construction we did "assign" a special imaginary unit for generalization from one algebra to another. I have heard of the words "simplex" and "perplex". Is there any widely accepted terminology?