Construction of dual numbers Out of curiosity, I was seeing about hypercomplex numbers. In that article, the definition says that, "Where possible, it is conventional to choose the basis so that $i_k^2 \in \{ -1, 0, +1 \}$. A technical approach to hypercomplex numbers directs attention first to those of dimension two.", where $i_k$ is the basis of hypercomplex number.
Later on, "Clifford algebraic construction" and "Cayley–Dickson construction" (along with "modified Cayley–Dickson construction") of hypercomplex numbers have been given. But both of these does not consider $i_k^2$ being $0$. Is there something that makes studying about dual numbers (separately) not much useful?
So, is there any other methods of construction which algorithmically produces dual numbers and its successors? Also, are there any methods in general that produce hypercomplex numbers like these?
 A: 
methods of construction

Dual numbers have a matrix representation (in fact more than one), or can be identified with $\Bbb R[X]/X^2$. Either shows duals are equiconsistent with reals. See here for a treatment of more general extensions of $\Bbb R$.

Is there something that makes studying about dual numbers (separately) not much useful?

Dual numbers and their generalizations have at least four applications beside their original motive, but you can't be blamed for not seeing them often. In principle, one can use square roots of $0,\,1,\,-1$ in an arbitrary sequence as one doubles the dimension, starting from $\Bbb R$. One can even vary anti/commutativity rules; for example, one can use $-1$ twice in more than one way, but they haven't been useful for equally many things.
In principle, an example of mathematics is useful insofar as it describes something we encounter, so why some examples are more useful than others amounts to asking why we encounter certain things, or even why the world is a certain way.
Philosophy aside, complex numbers have the advantage over split-complex numbers that they avoid zero divisors so are invertible, while dual numbers have the disadvantage relative to both that, because they're nilpotent, sums/differences of squares don't need them, because $(a\epsilon)^2=0$ just disappears. Mind you, sometimes that's useful.

Clifford algebraic construction and Cayley–Dickson construction (along with modified Cayley–Dickson construction) of hypercomplex numbers have been given. But both of these does not consider $i_k^2$ being $0$.

One modification of CD is to use a square root of $0$ rather than $-1$ to double the dimension. Although dual numbers can be related to a suitable Clifford algebra, most applications of such algebras warrant their being related instead to complex or split-complex numbers.
