Why are every structures I study based on Real number?

I've been studying basic concepts of inner product vector space, normed vector space and metric space. And all the inner products, norms and metrics are defined to be real-valued functions in my textbook. I doubted about it: why do I have to restrict the measurement standard to real number? Why not any ordered field? Why not… anything else?

I do know that there can be a lot of different ways to define 'distance': I'm studying topology, too. I'm just wondering if there is any more general definition to the inner product space, normed space, or metric space(my definition is just the standard one being taught at undergraduate level: only real-valued ones as I've mentioned).

For example, let $V$ be an $F$-vector space and $F_s$ be an ordered field embedded into $F$. Then I defined like this:

1. $\langle v, v\rangle=0$ if and only if $v$ is the $0$ vector
2. $\langle v, v\rangle>0$ (so the value is in $F_s$)
3. $\langle av, u\rangle=a\langle v, u\rangle$ where $a\in F$
4. $\langle v+u, z\rangle=\langle v, z\rangle+\langle u, z\rangle$
5. $\langle v, u\rangle=\langle u, v\rangle$

We can define the normed vector space in this fashion as well. I think this is a more general definition: I couldn't find anything wrong about it. I mean, if we define $\|v\|=\langle v, v\rangle^{1/2}$ then indeed $(V, \|\,\|)$ becomes a normed vector space according to my definition, as it should in the ordinary definition (I think the Cauchy-Schwarz inequality which connects the two spaces also hold in this general definition: I've checked the proof in my textbook and it does not use any property of the real number).

Is my definition not right, or not useful? If so, then why? I know that the real number is the only (up to isomorphism) ordered field with the least upper bound property and the property that every increasing bounded sequence converges. But is that enough to justify that every metric spaces use real number? I want to be more convinced, then.

A little bit disorganized, but I just wanted to hear some other people's opinion about this. Thanks as always.

P.S. I also know about the complex inner product vector space. The inner product there is complex-valued, so I think it was the start of my questioning.