Is there any example of function with high condition number? What is an  example of such a function in 2D and in high dimension space? 
 
This image is from CS231n stanford course. 
 A: Build a quadratic form using a diagonal matrix as following:
$$ {x}^{T} A x + {b}^{T} x + c $$
Where the matrix is given by:
$$ A = \begin{bmatrix}
1000 & 0 \\ 
0 & 0.0001
\end{bmatrix} $$
Then the condition number is:
$$ \operatorname{cond} \left( A \right) = \frac{ {A}_{11} } { {A}_{22} } = 1e6 $$
You can even get it higher if you want.
A: I always use the sillies functions when visualizing the multivariable concepts to build intuition. So, my favorite is $f(x,z)=x^2+az^2$. In this case it also roughly corresponds to the picture in your question if you take it that the horizontal axis is x, and the vertical is z. Here's the contour plot for a=2, which is similar to your picture:

Let's take its Hessian:
$$H_{xx}=\partial^2 f/\partial x^2=2$$
$$H_{xz}=\partial^2 f/\partial x\partial z=0$$
$$H_{zx}=\partial^2 f/\partial x\partial z=0$$
$$H_{zz}=\partial^2 f/\partial z^2=2a$$
Now your condition number is whatever you want it to be:
$$H_{zz}/H_{xx}=a$$
here's the plots for a=1 and a=1000:


This should give you an idea why it's so difficult to deal with high condition numbers. In a=1000 case the minimum looks more like a valley, it's hard to find its lowest point; while at a=1 it's a hole, and the bottom is easy to spot.
Imagine that you toss a steel ball inside these cavities. It'll settle at the bottom much faster in a=1. That's what drawing in your question tries to illustrate. 
