If your data lies $n$-dimensional space, hyperplanes will be $n-1$ dimensional. That is why in 1D it is point, in 2D it is line, in 3D it is plane etc., as pointed out in the comments. So, you should think accordingly and rephrase your question again.
A general equation for hyperplanes is $w^Tx+b=0$. For 1-D, we'll have $wx+b=0 \rightarrow x=-b/w$, a point. In 2D, this becomes $w_1x_1+w_2x_2=b$, which is a line equation. If you are seeking for a 3D hyperplane, which is a 3D shape with non-zero/infinite volume, it can easily be defined with four coordinates as follows: $w_1x_1+w_2x_2+w_3x_3+w_4x_4=b$, since the data lies in 4D.
I think, trying to visualize it directly in 3D doesn't make sense, because $n$ dimensional hyperplanes are visualized in $n+1$ dimensions. For example, you don't visualize lines in 1D, and planes in 2D. Line borders are meaningful inside a 2D plane, and plane borders are meaningful inside a 3D space. However, you can visualize some cuts through the 4D space, e.g. letting $x_4=c$, visualizing $w_1x_1+w_2x_2+w_3x_3=b-w_4c$ and by changing $x_4$, you could trace how this hyper-plane move through the 4D space. If you, for example, think the fourth dimension as time, you could trace your hyperplane's movements through time inside a video.