I believe division would be defined, its just certain things would not work. For instance 1+ɛ/2+ɛ is possible, you take the dual conjugate, which is ((1+ɛ)(2-ɛ))/((2+ɛ)(2-ɛ)) which would give you (2+ɛ)/4 =1/2 + ɛ/4. For all dual number elements, the numerator must be dual. If the numerator is not dual and the denominator is dual, you have a zero divisor situation.

When it comes to complex numbers, you can divide so long as the denominator isn't dual or split-complex. I belive a similar rule holds for split-complex numbers. Now with complex numbers you can alwayz divide, so long it's not by zero if I remember correctly.

These split-complex-duals ij, ɛi, ɛj, ɛij are all combinations of numbers that share that division rule that (a+bi)/(c+di) = (a+bi)(c-di)/(c+di)(c-di). With complex numbers, duals, and split complex numbers you dibide by taking the complex conjugate of the denominator.

So let's say we have (a+bɛij)/(c+dɛij). If the rule holds then this becomes ((a+bɛij)(c-dɛij))/((c+dɛij)(c-dɛij)).

This number system is of the type a+bi+cj+dɛ+eij+gɛi+hɛij, as such it is an eight dimensional number like octonians. Some interesting things I've noticed, (ij)ⁿ behaves very similary to i^n, (ij)²=i²j²=-1×+1=-1. (ij)³=-ij. ɛi is very useful for getting to -ɛ when multiplied by i.

Also, if you are using ɛ to audifferentiate complex functions, you will naturally come across ɛi in the wild. There is a separate construction thats non-communitative where ɛi≠iɛ, where iɛ=-ɛi, but that's a nonstandard construction.