Name of Octonions With Biquaternion Coefficients? Ordinary biquaternions are quaternions $(\mathbb{H})$ whose coefficients are complex $(\mathbb{C})$.  What is the name, analogous to "biquaternions", for octonions $(\mathbb{O})$ whose coefficients are ordinary biquaternions?  
If there is no such name, what is the most concise descriptive phrase?

The two things you are talking about are $\mathbb H \otimes \mathbb C$ and $\mathbb O \otimes (\mathbb H \otimes \mathbb C)$ (tensor products over $\mathbb R$).  I would call the former the complexified quaternions or the complex quaternion algebra (there is a unique quaternion algebra over $\mathbb C$ up to isomorphism, namely $M_2(\mathbb C)$.  (I have never heard the term "biquaternion" for this, and I am reasonably well acquainted with modern terminology regarding quaternion algebras, though I have not read the old works of Hamilton, Cayley, etc.)  
By analogy, I would call $\mathbb O \otimes (\mathbb H \otimes \mathbb C)
\simeq \mathbb O \otimes M_2(\mathbb C)$ something like the complexified quaternionic octonions, or possibly the two-by-two complex octonionic matrices, depending on how I represented them.  (The latter would be as $M_2(\mathbb O_{\mathbb C})$, where $\mathbb O_{\mathbb C} = \mathbb O \otimes \mathbb C$
is the (split) octonion algebra over $\mathbb C$.  Such tensor products have been examined
(in greater generality) in this paper, but I did not see a specific name for these objects, so I would venture that there is no name in common modern usage. (Admittedly, I just skimmed that paper quickly, so you might want to take a closer look.)
