How does a 3D, 4D,.. hyperplane look like (visualization)? I am looking at support vector machine classification algorithm.

It finds the optimal hyperplane. In linear algebra, hyperplane is a space that is one dimension lower than the ambient plane. For example, in a 2D space, the hyperplane is a 1D line. In a 3D space, the hyperplane is a 2D plane. The following image shows such examples.

I am interested to see the visualization of the hyperplane beyond 2D. For example: 3D, 4D...
As you can see from your examples a hyperplane of dimension $n$ is visualized in $n+1$ dimensional space. This goes basically back to the definition of hyperplane: 


  In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space. If a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperplanes are the 1-dimensional lines. This notion can be used in any general space in which the concept of the dimension of a subspace is defined.


Source: Wikipedia

Since we exist in 3D space (or 4D spacetime if you include time) one cannot visualize anything directly in 4D or higher. So you can't visualize a hyperplane with more than 2 dimensions directly either.