TITLE: How does quater-imaginary (and other imaginary/complex bases) work?
QUESTION [16 upvotes]: So I've been working on a simple base-conversion program, and having given it the ability to convert from decimal to any base $> 1$ or $< 0$, as well as the $p$-adic (bijective, I think?) bases, I decided to tackle imaginary systems.
Unfortunately, I can't seem to understand any of the articles I find about them.  The Wikipedia article on quater-imaginary is fairly comprehensible (I feel like I could write an algorithm for converting between decimal and it), but there are a couple of things I don't quite understand (why there are 4 digits, for example).
The article on complex bases is a bit more dense.  This is mostly because I know very little of the notation used (and trying to Google it proves difficult, since they tend to be symbols more than words).  I think I could extrapolate pure imaginary bases from quater-imaginary (once I've a fuller understanding of it), but I have no idea how to work complex bases at all.
So I guess my question is:  Does anyone have any good resources on how complex number systems work?  (That is to say, algorithms for converting between decimal and them, why some of the seemingly arbitrary decisions (like the number of digits) were made, et cetera).  Not necessarily in simple terms, but at least presented in such a way that I could Google or otherwise find the meaning of any of the things I didn't immediately understand without too much trouble.  (I'd rather be spending my time trying to learn about something than trying to search for information about it).

REPLY [4 votes]: I have worked on complex based systems ~2005 (also inspired me to asymmetric numeral systems in 2006) - just found this question and can clarify something if needed.
Some my materials: paper, slides, 3 interactive demonstrations, fractal Haar wavelets.
So basically it is just standard positional numeral system, but with complex base: you represent a complex number as $\sum_i a_i z^i$, where $a_i \in \{0,\ldots,n-1\}$. In 1D we have just base $z=\pm n$, in 2D we need $|z|^2=n$, in 3D $|z|^3=n$ and so on, but additionally the argument needs also to fulfill some conditions. I had a hypothesis (still unproven?) that only periodic cases worked well (subjection to $\mathbb{C}$, injection in all but a zero measure set): such that $z^2\in \mathbb{Z}z+\mathbb{Z}$. There is still infinite number of such cases - you can find their complete classification in my materials, also analytically found properties: area of fractional part (which is fractal), Hausdorff dimension of its boundary, circumference and area of convex hull.
The simplest nontrivial cases are $z=1\pm i$, which can be easily drawn on a graph-ruled notebook, e.g:

Regarding conversion, see slide 11, it is useful to write e.g. $2-2z+z^2=0$ for $z=1-i$ as $(2,-2,1)_z=0$. This way to convert e.g. $-4$ to  complex base:
$$(-3)_z\to^{+2} (1,-4,2)_z \to^{+2} (1,0,-2,2)_z \to^{+1} (1,0,0,0,1)_z $$
getting $-3=1+z^4$. Note that digit direction is reversed here: from the least to the most significant digit.