Verified answer

Writing z = x + iy, we have

|z - 3i| = |z - i|

==> √[x^2 + (y - 3)^2] = √[x^2 + (y - 1)^2]

==> x^2 + (y - 3)^2 = x^2 + (y - 1)^2

==> (y - 3)^2 = (y - 1)^2

==> y^2 - 6y + 9 = y^2 - 2y + 1

==> y = 2, a horizontal line.

I hope this helps!