An inner product is a symmetric positive definite bilinear form. In general fields, you'll happily be able to satisfy symmetric bilinear, but you'll struggle with positive definite: over a field of nonzero characteristic, you will not even be able to make sense of $\langle x, x \rangle \ge 0$, much less find a form for which it holds, since there is no ordering compatible with the field. Note that there is also no ordering on the complex numbers that is compatible with the field, but there is the subfield $\mathbb R$ which can be ordered, and so conjugate symmetry rescues us by ensuring $\langle x, x \rangle$ falls inside there, but fields of characteristic $p$ must contain a subfield $\mathbb F_p$ which is already not orderable.
Over fields of characteristic zero, like the rationals, you can find a reasonable inner product, but it's not as useful as you'd expect. For example, you can't get an orthonormal basis from a given basis, because you can't do square roots: you can find the norm squared, but not the norm itself. You could go all the way to the algebraics, or just as far as all the square roots, but at this stage I think you gain very little generality over just using $\mathbb R$ in the first place, and if you find you want completeness at any point, you'll be forced into the reals anyway.
I think I have too found the asymmetry between complex and real inner products to be frustrating. It's possible that there's a unifying theory that I'm unaware of, but I don't think it's unreasonable to view them as basically separate (if highly similar) entities, albeit entities that embed one in the other in a neat way. Essentially, $\mathbb R$ is special: it is, after all, the unique complete totally ordered Archimedean field, so it's not that surprising that we should pay it specific attention.
Isomorphisms are defined in many different contexts; but, they all share a common thread.
Given two objects $G$ and $H$ (which are of the same type; maybe groups, or rings, or vector spaces... etc.), an isomorphism from $G$ to $H$ is a bijection $\phi:G\rightarrow H$ which, in some sense, respects the structure of the objects. In other words, they basically identify the two objects as actually being the same object, after renaming of the elements.
In the example that you mention (vector spaces), an isomorphism between $V$ and $W$ is a bijection $\phi:V\rightarrow W$ which respects scalar multiplication, in that $\phi(\alpha\vec{v})=\alpha\phi(\vec{v})$ for all $\vec{v}\in V$ and $\alpha\in K$, and also respects addition in that $\phi(\vec{v}+\vec{u})=\phi(\vec{v})+\phi(\vec{u})$ for all $\vec{v},\vec{u}\in V$. (Here, we've assumed that $V$ and $W$ are both vector spaces over the same base field $K$.)
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So first of all, "integer" would not be adequate; vector spaces have fields of scalars and the integers are not a field. "Number" would be adequate in the common cases (where the field is $\mathbb{R}$ or $\mathbb{C}$ or some other subfield of $\mathbb{C}$), but even in those cases, "scalar" is better for the following reason. We can identify $c$ in the base field with the function $*_c : V \to V,*_c(v)=cv$. Especially when the field is $\mathbb{R}$, you can see that geometrically, this function acts on the space by "scaling" a vector (stretching or contracting it and possibly reflecting it). Thus the role of the scalars is to scale the vectors, and the word "scalar" hints us toward this way of thinking about it.