We introduce quaternary modified four $μ$-circulant codes as a modification of four circulant codes. We give basic properties of quaternary modified four $μ$-circulant Hermitian self-dual codes. We also construct quaternary modified four $μ$-circulant Hermitian self-dual codes having large minimum weights. Two quaternary Hermitian self-dual $[56,28,16]$ codes are constructed for the first time. These codes improve the previously known lower bound on the largest minimum weight among all quaternary (linear) $[56,28]$ codes. In addition, these codes imply the existence of a quantum $[[56,0,16]]$ code.