In addition to representing quaternions in the usual way they can be represented as a 2x2 matrix of complex numbers or as 4x4 matrix of real numbers:

representation 
linear combination 
quaternions 
w + x i + y j + z k 

2x2 matrix of complex 
w+y√1 
x+z√1 
x+z√1 
wy√1 


4x4 matrix of real 
w 
x 
y 
z 
x 
w 
z 
y 
y 
z 
w 
x 
z 
y 
x 
w 

w 
1 
0 
0 
0 
0 
1 
0 
0 
0 
0 
1 
0 
0 
0 
0 
1 

+x 
0 
0 
1 
0 
0 
0 
0 
1 
1 
0 
0 
0 
0 
1 
0 
0 

+y 
0 
1 
0 
0 
1 
0 
0 
0 
0 
0 
0 
1 
0 
0 
1 
0 

+z 
0 
0 
0 
1 
0 
0 
1 
0 
0 
1 
0 
0 
1 
0 
0 
0 


So we are replacing the i,j and k used in quaternions with the matrices:
These additional ways to code quaternions using matrices are not as efficient in terms of the number of variables needed to hold a given quaternion or possibly in terms of computation time. However there may be advantages in terms of getting an intuitive understanding of quaternions and beyond, especially if someone is familiar with linear algebra in terms of matrices.
So how do these notations compare with the usual way to represent quaternions as matrices?
1  2*y^{2}  2*z^{2} 
2*x*y  2*z*w 
2*x*z + 2*y*w 
2*x*y + 2*z*w 
1  2*x^{2}  2*z^{2} 
2*y*z  2*x*w 
2*x*z  2*y*w 
2*y*z + 2*x*w 
1  2*x^{2}  2*y^{2} 
The difference is that Pauli matrices translate points using the 'sandwich' product, in the same way as quaternions, as follows:
P_{out} = q * P_{in} * conj(q)
Whereas the traditional matrix transform has this form:
P_{out} = [M] P _{in}
The traditional matrix is not linear but is quadratic (terms like y^{2} and y*w), whereas their transform is linear, in contrast the Pauli matrix is linear but the transform is not linear.
Conventional Pauli Matrices
Although the above values work as required for example matricies representing i,j and k such as: 


square to 1. I have so far misrepresented the term Pauli matrices. What they really are is:

representation 
linear combination 
2x2 matrix of complex 
w + iz 
x  iy 
x + iy 
w  iz 


4x4 matrix of real 
w 
x 
y 
z 
x 
w 
z 
y 
y 
z 
w 
x 
z 
y 
x 
w 

w 
1 
0 
0 
0 
0 
1 
0 
0 
0 
0 
1 
0 
0 
0 
0 
1 

+x 
0 
1 
0 
0 
1 
0 
0 
0 
0 
0 
0 
1 
0 
0 
1 
0 

+y 
0 
0 
1 
0 
0 
0 
0 
1 
1 
0 
0 
0 
0 
1 
0 
0 

+z 
0 
0 
0 
1 
0 
0 
1 
0 
0 
1 
0 
0 
1 
0 
0 
0 


I think Pauli matrices are interesting though, because instead of having to learn the rules for multiplying the operators i, j and k these multiplication rules come automatically provided you know how to multiply 2x2 matrices. If you want to calculate the spin of fundamental particles in many dimensions then Pauli matrices may be the only way to do it.
The matrix is generated by multiplying the three imaginary values by the following Pauli matrices:
The real value is multiplied by the identity matrix.
The Pauli matrices in this form are not the exact equivalent of quaternions this is because, if we square them, we get +1 and not 1. In other words the identity matrix:
e1^{2} = 

* 

= 

e2^{2} = 

* 

= 

e3^{2} = 

* 

= 

So these dimensions are given by √+1instead of √1which quaternions use. In this form Pauli matrices have different properties, they don't form a normed division algebra. However we can convert Pauli matrices to have exactly the same properties to quaternions by multiplying by i. Then each of the generators are √1 as shown here:
e1^{2} = 

* 

= 

e2^{2} = 

* 

= 

e3^{2} = 

* 

= 

History
Pauli matrices were developed for physics (quantum mechanics) so that may be why they are formulated with this i factor difference from quaternions.
In quantum mechanics a + i b + j c + k d and a  i b  j c  k d
represent different spins for particles, so a particle has to rotate through
720 degrees instead of 360 degrees to get back where it started. Both Pauli matrices and quaternions have this property, see Spinors.
Generalizing Pauli Matrices
So far we have taken the Pauli matrices as a given but,
 How can we derive the Pauli matrices?
 Can we derive matrix representations of other algebras?
Since Pauli matrices square to 1 (the identity matrix) then one possible way to derive Pauli matrices is to find the square root of the identity matrix (square root of matrix discussed here).
To find a more general way to find matrices which are equivalent to other algebras we need to look at representation theory.
See also representation of Clifford algebra.