Rigorous meaning of the expression $dz = dx + idy$ Many complex analysis books just define $dz$ by $dz = dx + idy$.
In smooth manifold theory, the expressions like $dx$, $dy$, $df$ have precise meaning: covector field.
My question is:
What is the precise meaning of the expression '$dz = dx + idy$'? Do we understand this expression in terms of linear functionals on tangent space?

This is not explained super well in the literature.
A complex $1$-form eats real tangent vectors, and spits out complex numbers, in a real linear way.
How does the complex differential work?
Let $\mathbb{C}$ be identified with $\mathbb{R}^2$ in the usual way.  Then a function $f:\mathbb{C} \to \mathbb{C}$ can be thought of as a function $\hat{f}:\mathbb{R}^2 \to \mathbb{R}^2$.  This function has a derivative, which is a map between tangent bundles $D\hat{f}: T\mathbb{R}^2 \to T\mathbb{R}^2$.  The tangent bundle $T\mathbb{R}^2$ can be thought of as consisting of points $(p,v)$ where $p$ is a basepoint, and $v$  a vector.  Let $\pi:T\mathbb{R}^2 \to \mathbb{R}^2$ be the projection $(p,v) \mapsto v$.  Now $\pi \circ D\hat{f} :T\mathbb{R}^2 \to \mathbb{R}^2$.  Finally we can reinterpret this map as $df = \pi \circ D\hat{f} : T\mathbb{C} \to \mathbb{C}$, where $T\mathbb{C}$ is the real tangent bundle to $\mathbb{C}$.
You can compute that $$dz( a\frac{\partial}{\partial x} + b\frac{\partial}{\partial y}) = a + bi$$ that $$dx( a\frac{\partial}{\partial x} + b\frac{\partial}{\partial y}) = a$$ and that $$dy( a\frac{\partial}{\partial x} + b\frac{\partial}{\partial y}) = b$$
so the claimed equality actually does hold.  This still does not address what $\frac{\partial}{\partial z}$ means, but that is really another story.
