Conditions for Schur decomposition and its generalization

If the characterisic polynomial factors in linear factors then the Jordan decomposition works as your triangular matrix.

If you have a similar triangular matrix then the characteristic polynomial of $M$ is the characteristic polynomial of $T$ which clearly factors into linear factors.

So, the criterion is exactly the same as for Jordan decomposition.

The similar triangular matrix is just a lazy variant of Jordan decomposition.

This is a thought. If you look at the construction of schur decomposition, at every step, one uses a new eigenvector to triangularize further and further (see here). So as long as the matrix has $n$ eigenvalues (distinct or repeated), I don't see any problem in extending schur decomposition to any field.