A complex function identity

Question

Let $f(z) = \frac{-z + \sqrt{z^2 - 4}}{2}$ be a function on the upper half plane. The claim is that for $z = a + ib$ and $|a| \leq 2$, then $$ Im(f(z)) \approx \sqrt{k +b} $$ where $k = ||a| - 2|$ and $\approx$ means ignoring constants. I have tried Taylor series and the identity $$Im(\sqrt{a+ib}) = \frac{b}{|b|} \sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}}$$. None of these techniques have yielded a simple solution.