Just making sure this is clear: I don't actually guarantee any new features for this. This is just something I'm doing in my spare time (and sometimes I have other stuff I want to do), and also I'm not an expert on the stuff here; I'm only going to work on this if I'm sufficiently interested, have time, etc. I don't have an issue with suggesting features, as long as you understand that I don't have an obligation to add them.
ε₀: It's been a while since I worked on the ordinal numbers part, but from what I can remember, I had trouble finding information about how exactly to do what I'm trying to do here (take an arbitrary expression and simplify it to some canonical form); like, the definitions weren't worded in a way that makes this easy, and there were examples, but not enough to cover all the cases. I figured out enough to be confident about expressions involving ω, but not enough to be confident about expressions involving ε₀. (I could look at it again at some point, but again, I don't guarantee anything.)
ג_n: Gimel? I'm not familiar with that notation.
real number imprecision: because computers can't actually perform exact calculations on real numbers. I included "real numbers" mostly for completeness, but everything involving real numbers is still doing calculations using floating point numbers. I did start a rewrite that would give more exact answers *sometimes*, but I never got around to finishing it.
another suggestion about octonions: since we're unifying i j in C and H, i think it's nicer to have k as "the one spanning octonions from quaternions" (exactly what cayley-dickson construction did), despite there's convention from quaternions that "ij=k". this way we could have i, j, ij, k, ik, jk, (ij)k as e1~e7. furthermore, it's ok to use ÿ as ij digraph that also avoids the necessity of parenthesis in (ij)k. then they become i, j, ÿ, k, ik, jk, ÿk.
bug report: entering 1/94906266, the denominator from this on, C, H jump out mysteriously...
and suggestion continued: for split complex (i'd rather call them hyperbolic), i think it's better to use "i'"=1 since it's completely different from i. also j'^2=ÿ'^2=1. and there's split quaternion using i' j or i j' together.... and (completely) hyperbolic quaternion i' j' ÿ' and so on... imo it's generally better this way.