This
LogicalExpand[
Reduce[
{{a,b,c,d},{e,f,g,h},{i,j,k,l},{m,n,o,p}}.{a0+a3,a1-I a2,a1+I a2,a0-a3}==
{a0,a1,a2,a3},
{a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p}
]]
quickly gives you five different solutions.
Formatted differently gives the same solutions
A={{a,b,c,d},{e,f,g,h},{i,j,k,l},{m,n,o,p}};
X={{a0+a3,a1-I a2},{a1+I a2,a0-a3}};
LogicalExpand[Reduce[A.(X//Flatten)=={a0, a1, a2, a3},Flatten[A]]]
But I cannot understand
I want to get the solutions that express a _ i (i=0,1,2,3) as the functions of elements of X _ ij, e.g
when your original problem was to find the 4x4 matrix A and you showed an example 4x4 solution of coefficients.