How do you find the branch points of a function?

For example, f(z) = log ((1+sqrt(1+z^2))/2. Find its branch points.

It says the answer is 0, +i, -i, and infinity, but I have no idea how they got that.

When I plug in z=0, I get f(0) = log 1 = 0. When I plug in +i and -i, I get f(+ i or -i) = log 1/2...which isn't zero. So I don't know where they got how to get branch points (I initially thought it was just when f(z) = 0). Now I notice that 1+z^2 = 0 for +i and -i...However I am so confused how to calculate branch points.