These lines can be written is a different way:
L1: z = (1 + i) + r(3 + 4i)
L2: z = (1 + i) + s(3 − 4i)
If you plot these two lines in the xy-plane (using the y-axis for the imaginary axis), you will see that L1 passes through the points (1,1) and (4,5) and the line L2 passes through (1,1) and (4,−3).
The angle bisector could be horizontal or vertical. It is not clear what is needed here.
If you take the angle bisector to be horizontal, it has equation:
ABh: z = (1 + i) + t.
This is a horizontal line passing through (1,1). You could take any two points on this line as the answer. For example,
(1,1) = 1 + i or (2,1) = 2 + i.
If the angle bisector is taken to be vertical, it has equation:
ABv: z = (1 + i) + ui = 1 + i(u + 1).
This is the vertical line passing through (1,1). Two points on THIS line would be (1,1) = 1 + i and (1,2) = 1 + 2i.
Again, I cannot be sure WHICH angle bisector this question demands, so I have included both of them.
If I have not answered your question sufficiently, please post a reply and tell me why.