Can scipy calculate (double) integrals with complex-valued integrands (real and imaginary parts in integrand)?
Yes.  Those integrals (I'll assume they're area integrals over a region in 2D space) can be calculated using an appropriate quadrature rule.
You can also use Green's theorem to convert them into contour integrals and use Gaussian quadrature to integrate along the path.
(Couldn't upload the picture showing the integral as I'm a new user.)