Split-complex numbers and dual numbers

[ForMyThunder]

I recently had a conversation with a friend (who for some reason likes number theory) about these split-complex numbers and dual numbers. I'm more into topology, so I've never heard of them and he brought up that the modulus (I've only heard this term used for complex numbers) of the split-complex numbers can be negative. I made the guess that the modulus was a norm. He said it wasn't and I asked him what exactly a modulus was and he said it was just some real-valued function.

So, in general what is the definition of a "modulus" that coincides its definitions on the reals, complex, dual, and split complex numbers?

 

[rasmhop]

As I have come across the term in these contexts the modulus isn't just "some" real-valued function but rather the function
$$\|z\| = z\overline{z}$$
where $\overline{z}$ denotes the conjugate, so:
$$\overline{a} = a\qquad\textrm{for real a}$$
$$\overline{a+bi} = a-bi\qquad\textrm{for complex a+bi}$$
$$\overline{a+bj} = a-bj\qquad\textrm{for split complex a+bj}$$
$$\overline{a+b\epsilon} = a-b\epsilon\qquad\textrm{for dual }a+b\epsilon$$
So in the split complex case we have
$$\|a+bj\| = (a+bj)(a-bj) = a^2 -b^2$$
which is not a norm, but it is multiplicative and a quadratic form. In the dual case we have
$$\|a+b\epsilon\| = (a+b\epsilon)(a-b\epsilon) = a^2$$