No, Hermitian matrices can be over p-adics too. You just need an inner product so you can define a matrix to be self-adjoint, and you can define that inner product over p-adics. And the result is kind of funny, because although the statement of the result doesn't make any reference to complex numbers nor to adjoints, the proofs do. I do wonder if the result is still valid over other fields.
There are other notations for the Hermitian transpose, such as the dagger or the H. The asterisk is particularly physicist, I think.
And you might have studied physicist-accented mathematics; a lot of the accent often carries over into PDEs, functional analysis and related fields.
In my experience (as a physicist), the dagger is much more common in physics literature to represent adjoint; A^* for adjoint is confusing because it could just mean complex conjugation in many contexts. Overlines are practically never used physics notation.
I figured the notational confusion between the Hermitian transpose and the complex conjugate is kind of intentional, since moving from one side to the other of an inner product both conjugates scalars and is the adjoint of matrices, plus AA* is "real squaring" for both matrices and scalars.
There are other notations for the Hermitian transpose, such as the dagger or the H. The asterisk is particularly physicist, I think.
And you might have studied physicist-accented mathematics; a lot of the accent often carries over into PDEs, functional analysis and related fields.
The \equiv usage is in D \equiv diag(...)