Suppose $\varphi$ is a smooth map on $\mathbb{C}$. For a function $f$, we define a $0$-form $\varphi^*f$ as $\varphi^*f=f\circ\varphi$. Also, $ \varphi^*\,dx=\frac{\partial\varphi_1}{\partial x}\,dx+\frac{\partial\varphi_1}{\partial y}\,dy, \qquad \varphi^*dy=\frac{\partial\varphi_2}{\partial x}\,dx+\frac{\partial\varphi_2}{\partial y}\,dy, $ where $\varphi_1$ is the $x$ component of $\varphi$ and $\varphi_2$ is the $y$ component of $\varphi$.
I'm curious, are there sensible analogous definitions for $\varphi^*dz$ and $\varphi^*d\bar{z}$ for the complex case?
I know $dz=dx+idy$, so considering it as a $1$ form I thought maybe \begin{align*} \varphi^*dz &= \varphi^*(dz+idy)\\ &= (\varphi^* 1)\varphi^* dx+(\varphi^* i)\varphi^*dy\\ &= \varphi^*dx+i\varphi^* dy \end{align*} but I think it's not right to equate a constant with a constant function. Likewise in the case of $dz=dx-idy$. What is the proper definition for the complex differential?