At the moment I'm treating $ij=ji$ as its own quantity and reducing it where possible, as in $(ij)^2 =-1$ and $e^{ij} = \cos(j) + i\sin(j) = \cosh(i) + j \sinh(i)$.
From the first identity seems like you can say $ij=\sqrt{-1}=i$, and this even agrees with $\cos(j) + i\sin(j)= \cos(1) + i\sin(1)$ and $\cosh(i) + j \sinh(i) = \cosh(i) + 1\sinh(i)$ unless I made a mistake, but it still seems like a strange definition.
So, Is there a way to represent $ij$ as a linear combination of $1$, $i$, and $j$, and does the definition I came up with here break anything I haven't noticed?
No. What you are talking about is tessarines, a 4-dimensional algebra that combines complex and split-complex numbers.
In that system $ij$ is a separate unit vector, similar to $i$ in that $(ij)^2=-1$. But not equal to $i$. It is irreducible.
The algebra of tessarines is commutative and associative.