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 $a+bi+cj+d\epsilon:i^2=-1,j^2=+1,\epsilon^2=0$?

Dual tessarines. Combination of complex and split-complex numbers is tessarines, and when we add dual unit we usually add "dual" to the algebra's name.
UPDATE
One can implement this in Mathematica using the following code:
$Post=(Normal[Series[#,{ε,0,1}]]/.j->{-1,1})/.{x_,y_}->(x+y)/2+(j(y-x))/2&;

Test:
Log[j + ε]

Output:
(I π)/2 + 1/2 j (-I π + 2 ε)

UPDATE2
The following code also works (and works better):
$Post=((#/.ε->0)+ε(D[#,ε]/.ε->0)/.j->{-1,1})/.{x_,y_}->(x+y)/2+(j(y-x))/2&;

