Clifford algebra over non-Archimedean field

Question

Usually the Clifford algebra is defined over the Reals $\mathbb{R}$ or the Complex $\mathbb{C}$ numbers. Can the definition be extended over non-Archimedean fields, such as the hyperreal numbers $\mathbb{R^*}$? I could not find reference on the question.

Answer 1

The definition of the Clifford algebra involves only first-order relations. Therefore the transfer principle is applicable and the definition goes over to hyperreal fields which are elementary extensions of the real field.

Answer 2

The Clifford algebra is defined for any quadratic module over a commutative ring. See any algebra book, e.g. Bourbaki, Algebra IX, §9.

Answer 3

Also see section 2.5 of the paper:

Ochsenius, Herminia; Olivos, Elena, A comprehensive survey of non-Archimedean analysis in Banach spaces over fields with an infinite rank valuation, Shamseddine, Khodr (ed.), Advances in ultrametric analysis. Selected papers based on the presentations at the 12th international conference on $p$-adic functional analysis, Winnipeg, MB, Canada, July 2--6, 2012. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-9142-1/pbk; 978-1-4704-1024-7/ebook). Contemporary Mathematics 596, 215-236 (2013). ZBL1321.32010.