There are the complex p-adic numbers.
But what is the p-adic analogue of the Cayley–Dickson construction?
Or more important: What is the p-adic analogue of the octonions?
It would be nice if the (unit)-multiplication table of such a p-adic analogue corresponds to the projective plane over the finite field with p elements.
Is there any interesting mathematical structure with such a multiplication table for a general p?
Is there any research work about this?



p-adic analogue of octonions
Defining and classifying the octonion algebras (composition algebras of dimension $8$) over fields $k$, or, in more sophisticated terms, computing the Galois cohomology set $H^1(k, G\_2)$, is the topic of the book *Octonions, Jordan Algebras and Exceptional Groups* by T. A. Springer & F. D. Veldkamp (2000, appropriately published by Springer). Among other things, after properly defining what this means, they explain (§1.10(vi), page 22) that over the $p$-adic fields, just like over the finite fields or totally imaginary number fields, every octonion algebra is split, meaning it is isomorphic to the obvious one.


[These notes by P. Gille](http://math.univ-lyon1.fr/homes-www/gille/prenotes/octonions_beamer.pdf) discussing the more general problem of the classification of octonion algebras over rings, might also be worth looking at.


