This is going to be a somewhat vague question, but I'll be happy if you indulge me.
Euclidean space $\mathbb{R}^n$ is equipped with a lot of nice (algebraic, metric, topological,...) structure and has many nice properties as such, regardless of the dimension $n$ (simple-connectedness, finite-dimensionality, abelianness, having a uniform lattice, being a $\sigma$-finite measure space, and the list goes on...) . It seems interesting to me to try and find geometric/algebraic objects with non-integer dimension while trying to preserve as many properties of classical euclidean space as we can.
Some thoughts: the dimension of a vector space is always an integer so let's forget about vector spaces altogether (but stay in $\mathbb{R}^n$ for some large $n$) and interpret "dimension" as "Hausdorff dimension" (this seems geometrically intuitive to me, though I'm aware that there are many other notions of dimension in $\mathbb{R}^n$ or general metric spaces). One way of thinking of $\mathbb{R}^n$ is as a connected, simply-connected locally compact Hausdorff topological group, so I wonder now whether for every $\alpha > 0$ there is some "canonical" model of a topological group (implicitly assumed to lie in $\mathbb{R}^n$ for some $n$) with these properties and with the additional property that it is $\alpha$-dimensional (in the Hausdorff dimension sense). In the case where $\alpha$ is a positive integer, it is obvious what this model should be.