Math people:
The title is the question. The reason I am asking is that I am trying to determine exactly what fields can be used for an inner product. I posed that question at https://mathoverflow.net/questions/129413/what-fields-can-be-used-for-an-inner-product-space : if you have a comment about that question, please post it there and not here, lest anyone be accused of cross-posting. I am posting this question here because I am certain that someone here will know the answer and I am afraid that if I post it at MathOverflow, someone will complain that it is not a "research-level question", which I suppose it isn't.
Wikipedia asserts that only $\mathbb{R}$ and $\mathbb{C}$ can be used as the basefield for an inner product, but their explanation is incomplete and self-contradictory. They use the term "distinguished automorphism" without defining it or providing a link for its definition. I have Googled the term and found nothing. I know what an automorphism of a field is, but not a distinguished automorphism.
I am not an algebraist. I took a semester of undergrad abstract algebra and have done a fair amount of reading on abstract algebra. I would prefer a basic definition that does not involve differential geometry, differential topology, Lie groups, etc. A reference in the form of a hyperlink or a book would be welcome.
A distinguished automorphism is one that you distinguish. That is, the data that you're providing in some construction is not a field $F$ but a pair $(F, \alpha)$ where $F$ is a field and $\alpha$ an automorphism of it. $\alpha$ is the "distinguished" automorphism. As a more basic example of this terminology, suppose $(X, x)$ is a set $X$ and an element $x \in X$ of $X$. We might call such pairs "sets with distinguished points."
(You can also use the quaternions to construct Hilbert spaces.)