Why the $2 \pi i$ in the Tate twist?

Question

I am reading here about Hodge structures and in (1.4) the Tate Hodge-structure $\mathbb{Z}(n)$ and the Tate twist $V(n)$ on a Hodge structure $V$ are defined. I understand that one might want to consider only morphisms between Hodge structures preserving the weight and therefore the twist allows one to define morphisms between structures of different weights by changing the weight formally. But why do we need the $(2 \pi i)^n$? Can't we just take $\mathbb{Z}(n):=\mathbb{Z}$ for all $n$ as $\mathbb{Z}-$modules and simply define the Hodge structures as $H^{-n,-n}(\mathbb{Z}(n)):=\mathbb{C}$ (and $0$ otherwise)? Do we use the $(2 \pi i)^n$ simply as an indicator to remind ourselves of what $n$ is, or does it play any deeper role in the theory?