NormalMatrixQ[a_List?MatrixQ] := Module[{b = Conjugate@Transpose@a},a.b === b.a]
UnitaryQ[m_List?MatrixQ] := (Conjugate@Transpose@m.m == IdentityMatrix@Length@m)

m = {{1, 2I, 3}, {3+4I, 5, I}};
m //MatrixForm
->
(1	2I	3
3+4I	5	I)

ConjugateTranspose[m] //MatrixForm
->
(1	3-4I
-2I	5
3	-I)

{HermitianMatrixQ@#, NormalMatrixQ@#, UnitaryQ@#}&@m
-> {False, False, False}