Taking a base $z$ positional numeral system with digits $a_k\in \{0,\ldots,n-1\}$:

$$s:\left\{(a_k)\in\{0,\ldots,n-1\}^{\mathbb{Z}}: \exists_K \forall_{k>K} \ a_k=0\right \}\to \sum_{k\in\mathbb{Z}} a_k z^k $$

we usually consider $z=n$ (or $z=-n$), making $s$ "nearly bijection" with $\mathbb{R}^+\cup\{0\}$ (or $\mathbb{R}$): surjection and injection on all but a zero measure set (there is countable number of values with double representation, e.g. $0.11111(1) = 1.00000(0)$ for binary system $n=z=2$).

It turns out that we can also get such nearly bijection (not injective only on zero measure set) in higher dimensions, requiring e.g. that $|z|^2=n$ in 2D or $||Z||^3=n$ in 3D. Here are examples of using first non-negative powers for complex base numeral systems (easy to draw on graph-ruled notebook): $n=2$, $z=1\pm i$, the right one further recreates $\mathbb{Z}^2$ lattice:

Fractional part: $F_z=\left\{\sum_{k<0}a_k z^k:\ a_k\in \{0,\ldots,n-1\}\right\}$ is a simple IFS fractal: fulfills $z F_z=F_z \cup (F_z+1)\cup\ldots\cup (F_z+n-1)$.

Integer part: $I_z=\left\{\sum_{k\geq 0}a_k z^k:\ a_k\in \{0,\ldots,n-1\right\},\ \exists_K \forall_{k>K}\ a_k=0\}$ fulfills $I_z =zI_z\cup(zI_z+1)\cup\ldots \cup (zI_z+n-1)$.

We would like surjectivity: $I_z+F_z=\mathbb{C}$ and injectivity on all but a zero measure set (boundaries of such fractals): $\mu(F_z \cap (F_z+1))=0$.

There remains a difficult question of choosing $z=\sqrt{n}e^{i\varphi}$ argument to get it. Working on it a long time ago, I have concluded (without proof) that we need $I_z=\mathbb{Z}z+\mathbb{Z}$, requiring:

$$\textrm{periodicity condition}: z^2 \in \mathbb{Z}z+\mathbb{Z} $$

There is infinite (countable) number of such cases (twindragon and tame-twindragon for $n=2$, more for higher $n$), and we can tell nearly everything about such fractals, including analytic formula for area of fractional part, Hausdorff dimension of its boundary, circumference and area of its convex hull - some my materials: paper, slides, 3 demonstrations, fractal Haar wavelats.

**The remaining question is how to prove (or disprove) that periodicity is required for nearly bijection (not injective on at most zero measure set)?**

Here are such fractional parts for $n=2$, $z=\sqrt{2} e^{i\varphi}$ and growing $\varphi\in[\pi/2, \pi)$, below are numerical results of their Hausdorff dimension - for surjectivity it has to be two. We can see the three marked surjective cases: rectangle, tame twindragon and twindragon. They are all periodic - the question is if it is true for any $n$? In other words: if there exists a non-periodic surjective case with $|z|^2=n$?

Simple Mathematica source to generate such fractals, $n$ and $\varphi$ as above, $d$ first digits:

```
d = 14; n = 2; phi = Pi/4.; z = Sqrt[n] Exp[I*phi]; pow = Table[ReIm[z^k], {k, 0, d - 1}];
Graphics[{Point[Table[IntegerDigits[x, n, d].pow, {x, 0, n^d - 1}]]}]
```