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 $f(z)=Ae^{itheta}$ where $z=ae^{iphi}$ as $A(a, phi)$ and $theta(a, phi)$, would it be possible to visualize the sameness of derivatives at the point in all directions?