Does the observation that we can raise finite dimensional vector spaces to the power of certain hyperbolic numbers teach us anything?

Question

($U^*$ denotes the dual space of $U$ whenever $U$ is a vector space. I'll work over $\mathbb{R}$ for simplicity)

Given a finite-dimensional vector space $V$ and a hyperbolic number $a + bj$ with $a,b \in \mathbb{N}$, lets define $$V^{a + bj} := V^a \times (V^b)^*.$$

This is the only such function satisfying $V^0 \cong \mathbb{R}, V^1 \cong 1, V^j \cong V^*, V^{x+y} \cong V^x \times V^y.$ As a nice bonus, we get $(V^x)^y \cong V^{xy}$, suggesting that the usual multiplication of hyperbolic numbers is the "correct" multiplication for this application.

Question. Does this give us anything / teach us anything?