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?
I don't believe there is any standard terminology for this particular setup. However, the maps you want to define are a type of projection. For instance, for the case of quaternions you could denote the projection onto $\mathbb C$ by $P_{\mathbb C}$ and the map onto the orthogonal complement by $P_{\mathbb C^\perp}$.
But if you do want to give them suggestive names, in analogy with the terms "real" and "imaginary part" for complex numbers, I would suggest "complex" and "quaternionic part" for the case of these projections for quaternions and "quaternionic" and "octonionic part" for the these projections for octonions. They're not exact analogies, but reasonably close.