If $x,y\in\mathbb R$, then$$\sin\bigl(2(x+yi)\bigr)=\overbrace{\sin(2x)\cosh(2y)}^{=u(x,y)}+\overbrace{\cos(2x)\sinh(2y)}^{=v(x,y)}i.$$It is easy to check that both partial derivatives of both functions $u$ and $v$ are continuous everywhere.
If $x,y\in\mathbb R$, then$$\sin\bigl(2(x+yi)\bigr)=\overbrace{\sin(2x)\cosh(2y)}^{=u(x,y)}+\overbrace{\cos(2x)\sinh(2y)}^{=v(x,y)}i.$$It is easy to check that both partial derivatives of both functions $u$ and $v$ are continuous everywhere.