 Orthogonality of regular real and imaginary level curves

For a holomorphic function $f=u+iv$, the Cauchy-Riemann equations give the displayed gradient identities. Where $f^{\prime }\ne 0$, both level sets are regular curves; their nonzero normal vectors are perpendicular, so their tangents are perpendicular as well. The qualification is necessary. At a critical point both gradients vanish and the zero level sets may have several tangent branches. For $f(z)=z^2$ at zero, $u=0$ consists of two diagonals while $v=0$ consists of the two coordinate axes; there is no general orthogonality assertion at that point.