Over $\mathbb C$, every matrix is similar to a Jordan matrix $J=D+N$ with $D$ diagonal, $N$ nilpotent, and $DN=ND$. Transporting back gives $A=S+K$ where $S$ is diagonalizable, $K$ nilpotent, and they commute. This commuting decomposition is unique; its components are polynomials in $A$.
Example: $\begin{bmatrix}3&1\\0&3\end{bmatrix}=3I+\begin{bmatrix}0&1\\0&0\end{bmatrix}$. The diagonalizable part in the original basis need not itself be diagonal.
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