Q:

Can we visualize the fact that a holomorphic function has the same derivative at a point in all directions? 

A holomorphic function $f$ has the same derivative at a point in all directions, but can we visualize this fact? 
I guess probably not, at least if we try to visualize $f$'s real and imaginary components as a function of real and imaginary component of $z$.
Since $f=u+iv$ has real and imaginary parts $u, v$ , and the derivative of each part has real and imaginary parts, if we plot only $u$ as a function of $(x+iy)$ or $(x, y)$ in $\mathbb{R}^3$, then this function doesn't have the same derivative at a point in all directions (for both the derivatives of $u$ and $v$ usually have real and imaginary parts, say $a_1+ib_1$, $a_2+ib_2$; their 'combination' $a_1+ib_1+i(a_2+ib_2)$ is the same in all directions but that doesn't imply each of them is. 
(from somehow a perspective of linear algebra the combination doesn't, like a quaternion, have four independent component but two, and after the combination, the two independent components of the two derivatives somehow intersect each other, or say swops, and so loses their 'individuality'; it's like mixing two bottles of sands and water and that we can't separate out sands in one bottle after the mixture.) 
That means we can't find a tangent plane to a point, which would suggest that sameness of derivatives at the point in all directions, of the surface $u(x,y)$ (and $v(x,y)$). 
But by using other coordinate systems or decompose $f$ differently, like plot $f(z)=Ae^{i\theta}$ where $z=ae^{i\phi}$ as $A(a, \phi)$ and $\theta(a, \phi)$, would it be possible to visualize the sameness of derivatives at the point in all directions? 

A:

You could visualize how the function distorts a grid laid onto its domain, like this: 
The fact that the derivative is the same in every direction corresponds to the fact that in a small neighborhood of some point, the transformation can be described by multiplication with the derivative, and multiplication by a complex number is a dilation-rotation.  So if we zoom into the grid, the distortion becomes only a rotation, followed by stretching the grid uniformly in some direction.  This preserves the angles within the grid.  As you can see above, the rectangular grid is distorted to become curved, but all the angles are still right angles.  The whole thing gets a bit weird-looking when the derivative is $0$, because grid lines will be smooshed together at such a point, but other than that, we can talk about the angles and notice that they are preserved.  In that case (derivative $\neq0$), they're also called locally conformal transformations.  That's just the technical name for angle-preserving maps.