What is the complex formula for dilation with scale factor 2 and rotation by 90
deg around the center 3+4i?

  z' = 2iz +11 -2i

Step-by-step explanation:

Dilation multiplies each point by the scale factor. Rotation by 90° CCW is equivalent to multiplication by i.

When the center is not the origin, the transformation is applied to the difference from the center, then the result is added to the center.

  z' = 2i(z -(3+4i)) + (3+4i)

Simplifying gives ...

  z' = 2iz -6i +8 +3 +4i

  z' = 2iz +11 -2i

Check

For example, consider the point 1 unit east of the center of dilation/rotation. That point is z = 4+4i. Applying the transformation moves this point to ...

  z' = 2i(4 +4i) +11 -2i = 8i -8 +11 -2i

  z' = 3 +6i . . . . . 2 units north of the center of dilation/rotation

This is where it is expected to be after dilation by a scale factor of 2 and rotation 90° CCW.