In general, maps over $W$ from $W$ to $X \times W$ are the same as maps from $W$ to $X$, by the universal property of products.  Here, $W = \operatorname{Spec} \mathbb{C}$.  I think one possible reason for the apparent incongruity is that lots of complex points of the base-changed space end up more generic after dropping to the original space, because they aren't defined over $\mathbb{Q}$.

Let's consider an explicit example of what happens when we compose a map from $W$ to $W \times X$ along the projection to $X$, when $X = \mathbb{A}^1$.  In the base change, we have ideals in $\mathbb{C}[x]$ like $(x-\pi)$ that aren't defined over the integers.  These ideals lie in the preimage of $(0) \subset \mathbb{Z}[x]$ under the base change map, since the intersection with the subring is zero.

In higher dimensions, we can have $\mathbb{C}$-points with transcendental coordinates that still satisfy algebraic equations, hence lie on subvarieties, and these will map to the respective generic points under projection.