Another way of phrasing this: are there any viable definitions of something which is norm-like but whose range is in a linearly ordered rig (for example) rather than $\mathbb{R}$?

I have searched a fair bit (including in fairly encyclopedic textbooks), but have come up empty handed as to why everyone just uses $\mathbb{R}$ and does not consider generalizations.

The underlying motivation comes from looking at various theories of mathematics from a minimalistic, "universal algebra" perspective. From that way of looking at things, as opposed to a more semantic perspective which focuses on applications of norms, it seems difficult to justify why norms must range over $\mathbb{R}$. But perhaps it really is important that the range be Dedekind complete -- which would justify this choice. But this is currently not apparent to me.