You can use the arctan method if you want, but if you do you just need to be a little careful with the quadrants. In which quadrants is it possible that arg(z-1) be Pi/4 radians greater than arg(z+1)? You need to draw up a quick Argand diagram to find out.
Try it out in the 1st quadrant (where x>0 & y>0). Pick a spot at random here, and plot a point & call it z. Then plot z+1 and z-1 (which are 1 unit to the left & right respectively of z). You'll see that arg(z-1) is always > arg(z+1) so this allows for the locus to exist here. Try the same thing in the 2nd quadrant and you'll see something similar, that arg(z-1) is always > than arg(z+1).
Now try it in the 3rd & 4th quadrants (where y<0), and you will see that arg(z-1) is always < than arg(z+1) hence making it impossible for the locus to exist here.
This locus does not extend into the region y<0. What about when y=0? Well for this circle locus we get (centre at (0,1) & radius sqrt(2)), at y=0, x = 1 or x= -1. So z=1 or z=-1. Therefore either z-1=0 or z+1=0. This will mean arg(z-1) + arg(z+1) will be undefined in both cases, since arg(0) is undefined.
So the answer to this locus question is the circle of radius sqrt(2) and centre (0,1), and ONLY the part of it above the x-axis (i.e. where y>0)
The moral of the story: If you want to assume that arg(z) = arctan Im(z)/Re(z), that's fine. However if you make this assumption, then you must check each quadrant at the end, since this assumption can give some extra unwanted extensions to the locus (e.g. the part of the circle where y<0). If you check and verify each quadrant at the end, then it should be fine.
