Is there a name for this type of matrix ($C + C^\dagger$ is negative definite)?

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Given $C = A + B$, where $A$ is a complex, normal matrix with all negative eigenvalues and $B$ is skew Hermitian, is there an accepted name for $C$?

It has the property that $C + C^\dagger$ is negative definite.

If there's no accepted term for this kind of matrix, can anyone suggest the term they (you) would use for it?

I refer to matrices of this type frequently in a paper I'm authoring and I need a concise term to describe them.