The complex function $$ f(z) = \sqrt{(z-(a+b)) (z-(b-a))(z-(a-b)) (z+(a+b))}, \quad a,b\in \mathbb{R}, \quad a,b >0, \quad a > b $$ has 2 branch cuts located between $z \in [-a - b,b - a ]$ and $ z \in [ a-b, a + b]$. For the choice $z = c + i\varepsilon$, and $\varepsilon \to 0^{+}$ and e.g. $a = 1.2,b = 0.8$, the real and imaginary parts in Mathematica look as
How one would rotate the branch cut so the imaginary part does not suffer from the jump?