How does the differential of a complex endomorphism of $\mathbb C^n$ act on $\frac{d}{dz^i}$ and $\frac{d}{d\bar{z}^i}$?

Question

I caught myself up in a seemingly simple question. Suppose $f: \mathbb C^n \rightarrow \mathbb C^n$ is a $\mathbb C$-linear map given by a matrix $A \in M_n(\mathbb C)$. I am trying to figure out how the differential of $A$ acts on the standard basis in the complexified tangent space $(T {\mathbb{C}}^n)_{\mathbb{C}}$ - this basis denoted $\{ \frac{d}{dz^i}, \frac{d}{d\bar{z}^i} \}_{i=\overline{1,n}}$, where $\frac{d}{dz^i} = \frac{d}{dx^i} + i \frac{d}{dy^i}$ if $(x_i, y_i)$ are the standard real coordinates in $\mathbb C^n$ ($z_i = x_i + iy_i$).

I reasoned as follows. First, we need to see $f$ as a function between differentiable manifolds, $\tilde f:\mathbb R^{2n} \rightarrow \mathbb R^{2n}$. Then $\tilde f$ has matrix $\tilde A \in M_{2n}(\mathbb R)$, where $\tilde A$ is obtained from $A$ by replacing each $A_{ij} \in \mathbb C$ with a $2 \times 2$ real matrix which represents the complex number $A_{ij}$. Then we know that the differential of $\tilde f$ is also given by $\tilde A: T \mathbb R^{2n} \rightarrow T \mathbb R^{2n}$.

Further, I considered the linear map induced between the complexified tangent spaces, which is also $\tilde A$, but seen as a matrix in $M_{2n}(\mathbb C)$.

Now, I reasoned that to obtain the action of the differential of the initial function $f$ on $\frac{d}{dz^i}$ and $\frac{d}{d\bar z^i}$, we must do a change of basis in the complexified tangent space $(T \mathbb R^{2n})\otimes \mathbb C$, from the $\mathbb C$-basis $\{ \frac{d}{dx^i}, \frac{d}{dy^i} \}$ to the basis $\{ \frac{d}{dz^i}, \frac{d}{d\bar{z}^i} \}$. I calculated the change of basis matrix to be:

$$ U = \begin{pmatrix} I_n & I_n \\ iI_n & -i I_n \end{pmatrix} $$

Now I am left with the very ugly matrix: $$ U^{-1} \tilde{A} U $$ which I have virtually no idea how to relate to the initial $n \times n$ matrix $A$ with complex entries.

So my question is two-fold:

1. Is there something wrong with my approach? Or maybe I calculated $U$ wrongly?
2. If not, is there any way to express $U^{-1} \tilde{A} U$ in a "nice" way? Or anyway, is there a way to easily see the action of the differential of $f$ on $\frac{d}{dz^i}$ and $\frac{d}{d\bar{z}^i}$?