I'm working on a program which calculates in the flowsnake base (2.5-√-0.75, with cyclotomic digits) and I've come up with some observations and questions about this base representation. The program also computes the easternmost point on Gosper Island. It doesn't yet output it in flowsnake base, but it'll look something like 0.11166655544443332221116665554443333..., where the average length of a run is a transcendental number close to 3.140275. I'm using seven digits, where 0 and 1 mean themselves, but 2 means 0.5+√-0.75, 4 means -1, and so on.
In this base, some numbers have two representations and some have three. The ones with three representations are all 1/3 the sum of three numbers with finite representation, and are therefore Eisenstein rational (the ratio of two Eisenstein integers) and algebraic. Numbers with two representations can be algebraic (e.g. 1/2 is both 0.165432165432... and 1.432165432165...). All numbers on the shore of Gosper Island have two or three representations. If the representation repeats, the number is Eisenstein rational, and therefore algebraic. If the representation does not repeat, such as the easternmost point, is the number necessarily transcendental?
An algebraic number isn't necessarily an Eisenstein rational. For example, i is not Eisenstein rational (and contrariwise, ω, which is represented as 3 in this base, isn't Gaussian rational), and it's representation is the nonrepeating 3.1612256200055622045453655120223365.... A rational number can have only one representation, such as 1/8, which is 0.062602420464... (it has infinitely many zeros, so only one representation).
ETA: The easternmost point, expressed in flowsnake base, is 0.166655544433332221116665554443332222111666... and 1.422311266155564453342231126615564445334223.... These numbers are equal, just as 0.999... and 1.000... in decimal are. The point is 0.5749186263504636+0.070524523454486195i in decimal, within double precision, though I had to change ...636 to ...634 to get the representation starting with 0.1666.