What is the combination of Complex, Split-Complex and Dual Numbers If $a+bi:i^2=-1$ is a complex number, $a+cj:j^2=+1$ is a split-complex number, and $a+d\epsilon:\epsilon^2=0$ is a dual number; what is the term for the combination $a+bi+cj+d\epsilon:i^2=-1,j^2=+1,\epsilon^2=0$?

You have basically rediscovered the ring of 2x2 matrices, at least if you don't care about commutativity. If you want to think of these in a "hypercomplex" way, they are also called the "split-quaternions". See here: https://en.wikipedia.org/wiki/Split-quaternion
Note that they are using a different basis $i, j, k$; in your terms, we can instead say that $\epsilon = i+k$. There are actually several different nilpotents, but anyway, this algebra has the complex, split-complex, and dual numbers as elements.
This algebra is isomorphic to the ring of 2x2 matrices; see the matrix representation of $i, j, k$ in that article.
On the other hand, this is not a commutative algebra. If you want a commutative algebra which has these as sub-algebras, you could instead look at what Anixx is calling the "dual tessarines" above, which would be an 8-dimensional algebra generated as a quotient of the polynomial ring $\Bbb R[i, j, \epsilon]/(i^2+1, j^2-1, \epsilon^2)$.
