How does a 3D, 4D,.. hyperplane look like (visualization)?
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.