The Cauchy Riemann Equations: what do they really mean?

Question: Geometrical Interpretation of Cauchy Riemann equations?

One might think that being differentiable on $\mathbb{R}^2$ is sufficient for differentiability on $\mathbb{C}$. But the Jacobian of an arbitrary such function doesn't have a natural complex number representation.

$$
\left[ {\begin{array}{*{20}{c}}
{\partial u/\partial x} & {\partial u/\partial y} \\
{\partial v/\partial x} & {\partial v/\partial y}
\end{array}} \right]
$$
Another way of putting this is that no complex-valued derivative (see below for an example, known as the Wirtinger derivative) you can define for an arbitrary function fully captures the local behaviour of the function that is represented by the Jacobian.

$$
\frac{df}{dz} = \left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \right) + i\left(\frac{\partial v}{\partial x}-\frac{\partial v}{\partial y}\right)
$$
The idea is that we should be able to define a complex-valued derivative "purely" for the value $z$, without considering directions, i.e. we want to consider $\mathbb{C}$ one-dimensional in some sense (the sense being "as a vector space"). More precisely, the derivative in some direction in $\mathbb{C}$ should determine the derivative in all other directions in a natural manner -- whereas on $\mathbb{R}^2$, the derivatives in *two* directions (i.e. the gradient) determines the directional derivatives in all directions.

If you think about it, this is quite a reasonable idea -- it's analogous to how not every linear transformation on $\mathbb{R}^2$ is a linear transformation on $\mathbb{C}$ -- only spiral transformations are.

$$
\left[ {\begin{array}{*{20}{c}}
{a} & {-b} \\
{b} & {a}
\end{array}} \right]
$$
How would we generalise differentiability to an arbitrary manifold? Here's an idea: a function is differentiable if it is locally a linear transformation. So on $\mathbb{R}^2$, any Jacobian matrix is a linear transformation. But on $\mathbb{C}$, only Jacobians of the above form are linear transformations -- i.e. the only linear transformation on $\mathbb{C}$ is multiplication by a complex number, i.e. a spiral/amplitwist. So a complex differentiable function is one that is locally an amplitwist (geometrically), which can be stated in terms of the components of the Jacobian as:

$$
\begin{align}
\frac{\partial u}{\partial x} & = \frac{\partial v}{\partial y} \\
\frac{\partial u}{\partial y} & = - \frac{\partial v}{\partial x} \\
\end{align}
$$
This is precisely why you shouldn't (and can't) view complex differentiability as some basic first-degree smoothness -- there is a much richer structure to these functions, and it's better to think of them via the transformations they have on grids.

One might observe that the Cauchy-Riemann equations can also be restated as: $\frac{\partial f(z)}{\partial\bar{z}}=0$, where the derivative is the Wirtinger derivative defined above. This seems completely bizarre to me, but apparently it makes sense with some differential geometry, and I'm given the keyword "complexifying the tangent bundle".