  For any complex number $\alpha$ and any even-size skew-symmetric matrix $B$,
we define a generalization $\pfa{\alpha}(B)$ of the pfaffian $\pf(B)$ which we
call the $\alpha$-pfaffian. The $\alpha$-pfaffian is a pfaffian analogue of the
$\alpha$-determinant. It gives the pfaffian at $\alpha=-1$. We give some
formulas for $\alpha$-pfaffians and study the positivity. Further we define
point processes determined by the $\alpha$-pfaffian. Also we provide a linear
algebraic proof of the explicit pfaffian expression for the correlation
function of the shifted Schur measure.
