 Two-sided cluster set at an interior slit point

In the explicit slit map, right-side approach to $i/2$ gives the positive root, and left-side approach gives the negative root. The corresponding lower-half-plane boundary values are $\pm 25/(2\sqrt{21})$, so the two unit-circle image values are distinct. Every other conformal bijection onto the disk differs by a disk automorphism, which extends bijectively to the unit circle. Thus a sequence approaching that slit point has exactly one or both of two boundary cluster values, according to which sides occur infinitely often.