Part b i showed in the picture, i think the first part seems ok, then showing for elsewhere i just let z0 be any point, and tried to find if there were any points besides zero where the limit exists but lim z0/delz is infinity, so is that sufficient to say that that its only differentiable at zero?

We did somthing different as well, let it be z= u+iv, then f(z) = u^2 + ivu cause Re(z) = u, then let these u^2 and vu be two new variables and applied C-R using implicite differentiation and found somthing different, that it is differentiable when Re(z) = constant or Im(z) = 0

Am i on the right track at all?