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The work follows:

let z=a+bi

|a+bi-1|=|a+bi+i|

(a-1)^2+b^2=a^2+(b+1)^2

a^2-2a+1+b^2=a^2+b^2+2b+1

-a=b

the correct answer should be a perpendicular bisector of segments joining z=1 and z=-i

my result looks more like a perpendicular bisector of segments joking a=0 and b=0

where did I go wrong? I've been confused about this for a while.