Questions like this can often be solved geometrically.
On the complex plane, | z - 1 | is the distance between the points z and (1, 0).
So, for example, if | z - 1 | = 1, the variable point z would be constrained to move on the circle centre (1, 0) with radius 1.
Here, | z - 1 | = | z + 3 |, says that z is equidistant from the points (1, 0) and (-3, 0) which means that it lies on the vertical line z = -1.
Similarly | z - 1 | = | z - i | says that z is equidistant from the points (1, 0) and (0, 1) meaning that it lies on the 45 deg line through the origin.
To satisfy both conditions, z must be at the intersection of these two lines, and that is easily seen to be the point (-1, -1).