For more information about double numbers see this page.
Algebra Laws
Double Numbers over the real numbers are a 'field' they have the following properties:
addition  multipication  

unit element  0  1 
commutative  yes  yes 
associative  yes  yes 
distributive over addition    yes 
inverse exists  yes  yes 
As an Extension Field to Real Numbers
R[x]/<x²1>
As a Multiplicative Group
If we ignore addition and treat complex numbers as a group then the group is equivalent to a D(2), it has the following properties:
Cayley Table
The Cayley table is symmetric about its leading diagonal:
1  D  1  i  
1  1  D  1  D 
i  D  1  D  1 
1  1  D  1  D 
i  D  1  D  1 
For more information about Cayley table see this page.
Cayley Graph
For more information about Cayley graph see this page.
Cyclic Notation
The group has two generators each cycling between two elements:
(1,D)(1,D)
(1,1)(D,D)
For more information about cyclic notation see this page.
Group Presentation
There is only one generator which when applied n times cycles back to the identity.
<a,b  a²=1,b²=1,ab=ba>
For more information about group presentation see this page.
Group Representation
A representation using 4 ×4 matricies containing 0 and 1 is:
[ 

] 
An alternative 2×2 matrix representation containing 0, 1 and 1 is:
[ 

, 

] 
For more information about group representation see this page.