For more information about complex numbers see this page.
Algebra Laws
Complex 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]/<x²+1>
As a Multiplicative Group
If we ignore addition and treat complex numbers as a group then the group is equivalent to a cylcic group of order 4, it has the following properties:
Cayley Table
The Cayley table is symmetric about its leading diagonal. For a cyclic group the table can be drawn with same terms on the bottomleft to topright diagonals:
1  i  1  i  
1  1  i  1  i 
i  i  1  i  1 
1  1  i  1  i 
i  i  1  i  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 is shown as a single cycle:
(1,i,1,i)
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.
<i  i ^{4 }=1>
For more information about group presentation see this page.
Group Representation
This is the 4^{th} root of the identity matix (such that lesser roots are not identity). See this page for information about taking roots of a matrix. One a matrix which will do this is an 4×4 matrix of this form:
[i] = 

An alternative 2×2 matrix using positive and negative reals is:
[i] = 

For more information about group representation see this page.