In 3 dimensions, there is a concept of rotating about an arbitrary line that is not centered at the origin. To pull this off, we first move the origin so that it is on this line. Anywhere on the line will do. Then, we do the rotation. And finally, move the origin back to its original location.
I'm thinking about the counterpart to this in 4 dimensions. Now, there is no such thing as rotation about a line. There is a 2-dimensional plane that is not affected by the rotation (this is the counterpart to the line in 3-d) and a 2-dimensional plane orthogonal to the first one in which the rotation happens.
If we were to extend the 3-d rotation to the 4-d case, then if we want to rotate about an arbitrary plane (not passing through the origin), we should first translate the space until the origin is somewhere in the 2-d sub-space that is not going to get impacted by the rotation. Then, we should do the rotation and finally, translate the origin back.
First, is this the right approach? And second, I'm just blindly extending what we happen to do in 3-d space (which there is visual intuition for) to the 4-d case. Can anyone shed light on the deeper reasons that extend to arbitrary dimensional spaces?
Here's a somewhat more algebraic way of thinking about rotations rigid motions in arbitrary dimension which will (hopefully) shed some light on your issue.
Some terminology
Note that the precise meaning of these terms vary from author to author. This is just how I've chosen to define them in this answer, for clarity.
By rotations (alternately special orthogonal transformations), I'll refer to linear maps $\mathbb{R}^n\to\mathbb{R}^n$ which preserve the dot product and the orientation. These are exactly the maps which correspond to orthogonal matrices with positive determinant (i.e. matrices $A$ satisfying $A^TA=I$, $\det(A)=1$). The set of all such rotations is called the special orthognal group $SO(n)$.
By rigid motion (alternately special Euclidean transformations), I'll refer to maps $\mathbb{R}^n\to\mathbb{R}^n$ (not necessarily linear) which preserve the Euclidean distance between all pairs of points and orientation. The set of all such motions is called the special Euclidean group $SE(n)$.
Some facts about $SO(n)$
By looking at the eigenspaces of a rotation $R\in SO(n)$, one can deduce that there are $\left\lfloor\frac n2\right\rfloor$ orthogonal planes (not necessarily unique) such that $R$ decomposes into a 2D rotation on each plane. Note that in odd dimensions there will be a line left over, which is fixed by $R$.
This gives a some special class of rotations: In any dimension, there are some rotations which rotate by an angle $\theta$ in one plane and fix all the others; such rotations are called simple. In $n=4$, there is also the special case in which the two rotation angles are the same; these are called isoclinic rotations. Generally, though, one cannot think of a rotation as "about an axis" or "about a plane"; the decomposition into rotation planes is the fundamental object. In $n=4$, for instance a non-simple rotation will not fix any plane, instead there will be two rotation planes and each is rotated "about" the other.
Some facts about $SE(n)$
Clearly, rotations are themselves rigid motions which preserve the origin. Additionally, we have the translations $T_v$ defined by $T_v(u)=u+v$ where $u,v\in\mathbb{R}^n$. It turns out that these two subgroups generate the entire special Euclidean group, in that every element can be written as $T_v\circ R$ for some rotation $R$ and some translation $T_v$; These have the nice composition law $(T_u\circ R)\circ (T_v\circ R')=T_{u+Rv}\circ(RR')$. A "rotation about a point $u$" can be thought of as translating $u$ to the origin, applying a rotation, and translating the origin back to $u$, i.e. $T_u\circ R\circ T_{-u}=T_{u-Ru}\circ R$.
Be warned though, that not every rigid motion is a rotation about some point. In addition to pure translations in dimension $n\ge 2$, we also have screw translations in dimension $n\ge 3$, where we apply a rotation fixing some axis and also translate parallel to that axis. In higher dimensions, this remains possible; if part of the translational part is in a direction fixed by the rotational part, then we cannot write the resulting transformation as $T_u\circ R\circ T_{-u}$. If what you're interested in is rigid motions and not merely "rotations about a point", then it may nor be best to think about the former in terms of the latter.