Chapter 6 of Michael Wolf's notes (MichaelWolf/QChannelLecture.pdf) discuss the structure of the spectrum of quantum maps and channels. However, it seems like the only explicit example given in the section is Example 6.1, which discusses the determinant of $T(\rho)=(\rho^{T_c}+I \operatorname{tr}(\rho))/(d+1)$.
One thing that can be used in this context, and is discussed in chapter 2 of the notes, is the relation between adjoint of the map and Hermitian conjugate of its matrix representation. Given a map $\Phi$, and denoting with $\hat\Phi$ its representation as a linear operator, we have $\Phi^\dagger=\hat\Phi^\dagger$, if $\Phi^\dagger$ is the adjoint map, defined as $\langle X,\Phi(Y)\rangle=\langle\Phi^\dagger(X),Y\rangle$ for all $X,Y$, and $\hat\Phi^\dagger$ the Hermitian conjugate taken with respect to some matrix representation of $\hat\Phi$. It follows that $\Phi$ has real eigenvalues iff it's self-dual. But if $\Phi(\rho)=\sum_a A_a \rho A_a^\dagger$ then $\Phi^\dagger(\rho)=\sum_a A_a^\dagger\rho A_a$. It follows that for any channel, $\Phi$ has real eigenvalues iff it has a Kraus decomposition in terms of Hermitian operators.
So a simple class of maps with nonreal eigenvalues are $\Phi(\rho)=A\rho A^\dagger$ with $A$ non-Hermitian. And in such cases $|u_i\rangle\!\langle u_j|$ is an eigenvector with eigenvalue $\lambda_i\bar\lambda_j$, if $A|u_i\rangle=\lambda_i|u_i\rangle$.
With this we can understand the spectrum of the simple class of maps $\rho\mapsto A\rho A^\dagger$. What are some other interesting examples where we can compute eigenvalues/eigenvectors?