Complex Derivative: Directional Derivatives & Complex Variables

[Skrew]

I'm not sure if it's OK to post this question here or not, the Calculus and Beyond section doesn't really look very heavily proof oriented.

I'm trying to prove that if continuous complex valued function f(z) is such that the directional derivatives(using numbers with unit length) preserve angles then the complex derivative exists.

Similarly I need to prove that if the directional derivatives have all the same norm values, then f(z) has a complex derivative.

So far I have proved that if the directional derivative preserves angles then difference quotient from every direction are all linear multiples of each other.

I need now to prove that the norms are the same, I don't know how to link up the norm length with the angles.

 

[fresh_42]

Angles and lengths are related by
$$
\cos \sphericalangle (\vec{a},\vec{b})=\dfrac{\vec{a}\cdot \vec{b}}{|\vec{a}||\vec{b}|}
$$

However, I'm not sure whether the result holds at all, especially how to define angles if there isn't already a directional derivative, and where the mixed terms of the Cauchy-Riemann conditions come into play.