  In this paper we consider a stronger property than the
Bishop-Phelps-Bollob\'{a}s property for various classes of operators on a
complex Hilbert space. The Bishop-Phelps-Bollob\'as {\it point} property for
some class $\mathcal{A} \subset \mathcal{L}(H)$ says that if one starts with a
norm one operator $T$ belonging to $\mathcal{A}$, which almost attains its norm
at some norm one vector $x_0$, then there is a new operator $S$, belonging to
the same class $\mathcal{A}$, which is close to $T$ and attains its norm at the
same vector $x_0$. We study it for classical operators on a complex Hilbert
spaces such as self-adjoint, anti-symmetric, unitary, compact, normal, and
Schatten-von Neumann operators. We also solve analogous problems by replacing
the norm of an operator by its numerical radius.
