I've encountered this operation a few times now and I don't really understand its meaning. Given a vector space $V $ we can "complexify" it by the operation $V\mapsto V\otimes \mathbb {C} $.
First of all, a sanity check - this tensor product is over the reals, right? Secondly, how do I imagine this complexified version and why is the tensor product with the complex plane the right way to do it? I do not need an extensive explanation about tensor analysis, just intuition on this apparatus.
This tensor product is over whatever field your vector space $V$ is over. So it's probably the reals, yes, but it might well also be the rationals or whatever.
Intuitively: it's the same thing, just that you're now allowed complex coefficients.
It might help you to try to prove the following: if $V$ is a vector space over $\mathbb{R}$ with basis $v_1, \dots, v_n$, then $V\otimes_\mathbb{R} \mathbb{C}$ is a vector space over $\mathbb{C}$ with basis $\hat{v}_1, \dots, \hat{v}_n$, where $\hat{v}_i = v_i\otimes 1$.
(Once you get comfortable with the object $V\otimes_\mathbb{R} \mathbb{C}$, you can safely abuse notation slightly and drop the hats.)