Complex Square root bound

Question

Under what conditions on $a, b$ and the branch cut of the square root does the following hold: $$ |\sqrt{a + b} - \sqrt{a} | \leq C \frac{|b|}{\sqrt{|a| + |b|}} $$ for a universal constant $C > 0$. This seems to be true in the real case, but I'm not familiar enough with the subtleties in the complex plane to generalize this.