Rotation via multiplication

Image. $A'$ has affix
$$z'=(1+i)e^{i\pi/3}=(1+i)\left(\frac12+i\frac{\sqrt3}{2}\right).$$

Compute.
$$z'=\frac12+i\frac{\sqrt3}{2}+\frac i2-\frac{\sqrt3}{2}=\frac{1-\sqrt3}{2}+i\frac{1+\sqrt3}{2}.$$

Conclusion. The image has affix $\frac{1-\sqrt3}{2}+i\frac{1+\sqrt3}{2}$.