Amazing question which has a long answer, but I will try to be concise. In the context of Krylov subspace methods for general matrices, the eigenvalues of a non-symmetric matrix mean very little. In “Any nonincreasing convergence curve is possible for GMRES”, Greenbaum et al. show that any nonincreasing convergence curve is possible for GMRES independent of the eigenvalue distribution of the matrix. To remedy this problem, Trefethen suggests pseudospectra of a matrix and pseudospectra-convergence analysis for various Krylov subspace methods (“Generalizing eigenvalue theorems to pseudospectra theorems” and “Pseudospectra and Spectra”, but really checkout anything Trefethen wrote on Krylov methods and pseudospectra). However, in some cases, determining pseudospectra of a matrix is difficult (without having the matrix). For example, if you are investigating the matrix representation of a linear operator (say, discretization of a PDE), you have a lot of functional analysis tools which don't help with the pseudospectra analysis at all.
Hence, other people are pushing for another analysis technique called numerical range (or, field of values). This technique is amenable to functional analysis techniques and there has been some exciting developments recently. (See “The Field of Values Bounds on Ideal GMRES” for example, but the last 10 or so years were amazing in terms of new theoretical results and their applications).
However, all of these methods provide an asymptotical limit, not a practical one. For example, you regularly see optimality (in the context of preconditioners) proofs, but to observe optimality in practice one has to be solving very large problems. Mark Embree (who worked with Trefethen on pseudospectra) has the paper “How descriptive are GMRES convergence bounds?”, where he demonstrates that all three types of bounds fail to provide informative and meaningful predictions regarding the convergence. They are usually incredibly pessimistic, and the convergence is achieved much earlier than predicted in practice.
I tend to use field of values in general, even for symmetric matrices -though you have to be careful with indefinite matrices, ratio field of values or other variants are better to use in that case, or use eigenvalues-. Most of the time, I interpret the result I obtained qualitatively rather than quantitatively. Which means that if I can prove that the residual reduces by 10% every iteration (miserably slow) independent of a parameter, I don't conclude that the convergence (tol < 1e-8) will be achieved in 175 iterations. I conclude that independent of the parameter of interest, the problem is solvable at most some number $k$ iterations which will be much smaller than the matrix size $n$.
There will be a lot of typos/grammatical mistakes in this answer, but it is late here. My apologies.