Why is complex derivative direction independent Visual complex analysis by Tristan Needham provides a great intuition by explaining the beauty of complex analysis. I have a question regarding the complex derivative.
The image above shows a function $w(z)$ from $\mathbb{C}$ to $\mathbb{C}$. When the complex number $z$, is moved to $z+dz_{1}$ or $z+dz_2$ the corresponding values are $w(z+dz_1)$ and $w(z+dz_2)$. Let's denote $dw_1=w(z+dz_1)-w(z)$ and similar thing with $dw_2$.
In that book complex derivative is defined, as a complex number such that
$$dw_1=w'(z)dz_1$$
$$dw_2=w'(z)dz_2$$....
My question:-
Now, I am unable to convince myself about why $dz_1$ and $dz_2$ are multiplied with the same complex number to get $dw_1$ and $dw_2$ respectively.
 A: This is really intereting! Your confusion reminds me of mine in college.
As @march mentioned, this is the definition of the complex derivative.
You can come up with a random differentiable mapping $f:\mathbb{R}^2\to\mathbb{R}^2, (x,y)\mapsto (u,v)$ and write it in terms of complex variable $z,\bar z$. Then it's very likely it will not suffice the condition above. A complex function differentiable in the sense of real function is not necessarily complex differentiable, which is a stronger property.
One simple example is $f(z)=\bar z$

Easy to see, you get different "directional derivative" when approaching from different directions. Then we say this function doesn't have a complex derivative.
All the functions that are complex differentiable are called holomorphic function.
The condition that is Cauchy Riemann equation or as a rule of thumb, your expression of complex function in terms of $f(z,\bar z)$ have zero partial derivative to $\bar z$ (or doesn't depend on $\bar z$.
If you are interested in why complex derivative is defined as strong as such... then it's a question of taste or word use.
A: I don't know the argument presented in your book, but hopefully this will help. The comment above is good. To add, notice that the definition of the derivative is the same as in the real case. There are no restrictions on the limiting variable h. Think about the formal definition of a limit, where the absolute value of h is used. There is a whole circle of points with the same modulus for any given h. This illuminates what we mean when we say the limit has to agree "in all directions."
A: I mean firstly, the most important thing to understand is that what you said not true for all types of mappings. It is only true for very special type of mapping known as Analytic mappings.
In-fact an often not mentioned point is that Complex differentiability is a much stronger condition than regular partial differentiability. You can find further details if you keep on reading the book. IF you have questions, feel free to comment on my post.
