Normally, the Pfaffian is defined for real skew-symmetric matrices $A = -A^T$, and some authors extend this to complex skew-symmetric matrices. Is there a straightforward generalization of the Pfaffian to complex skew-Hermitian matrices $A = -A^* \ne A^T$, however?
(Note that $\sqrt{\det A}$ is not sufficient because it doesn't tell you the sign.)