The difference between the arguments of two complex numbers is the argument of their quotient. So . You want that argument to be .
Now a complex number has argument if (and ).
So let . Then . Find the real and imaginary parts of that number in the usual way (by rationalising the denominator) and write down the condition that the real part is the negative of the imaginary part. That will give you the equation of a circle. If you want extra credit, you could point out that the locus is in fact only part of that circle (because of the condition in the previous paragraph, which says that the imaginary part of should be positive).