I am seeking a general solution to the initial value problem x' = Ax, x(0) = x_0 that can be written out to include both the eigenvalues and eigenvectors.
To cover the case of repeated eigenvalues, we can certainly come up with this:
x_(t) = c_1*x_1(t) + ... + c_n*x_n(t)
where x_i(t) is given by this procedure I found:
However, this does not cover the case of complex eigenvalues. (Does it?) Many differential equation books cover the topic of complex and repeated eigenvalues of matrices. However, is it possible to have a real valued nxn matrix with complex eigenvalues that are also repeated? It seems likely, but I can't find a single statement on it one way or the other.
And if so, can we just stick of sin/cos on the end of that equation above to give the most general solution, or would repeated complex values mess that up?
Yes, that procedure covers the case of complex eigenvalues just fine. What it does is compute the exponential of $At$ by writing it in Jordan normal form, which you can find explained in most textbooks on linear algebra.
It is possible to have a real $n \times n$ matrix with repeated complex eigenvalues, with geometric multiplicity greater than $1$. You can take the companion matrix of any real monic polynomial with repeated complex roots. The smallest $n$ for which this happens is $n = 4$. For example, taking the polynomial $(t^2 + 1)^2 = t^4 + 2t^2 + 1$ gives the matrix
$$M = \left[ \begin{array}{cccc} 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & 0. \end{array} \right].$$