In This question I tried to solve this problem
Show that as $z$ traverses a small circle in the complex plane in the positive (counterclockwise) direction, the corresponding point $P$ on the sphere traverses a small circle in the negative (clockwise) direction with respect to someone standing at the center of the circle and with body outside the sphere.
I realised I have no idea how to express a rotation or rotation direction in $\mathbb{R} ^n$ mathematically. It is easy to do that in the complex plane > $a+bi + re^{i \theta} $ detriment a rotation for a point moving in complex plane with center at $a+bi$ and radius $r$ if $\theta$ increases from $0$ to $2\pi - \varepsilon $ this rotation is positive otherwise the rotation is negative
It is obvious that a rotation of a point around other point is determined by the two points, the plane that contain the rotation in $\mathbb{R} ^n$ and the direction of rotation.
This is equivalent to finding an intersection of an $n$ dimension sphere and a 2D plane. But how to express the rotation direction mathematically? I don't remember anything from analytic geometry so I tried to see in my old calculus books which contained some sections on analytic geometry but I didn't find anything useful.
If you want something that appears like your complex number formula, you can use the fact that for a skew symmetric matrix $A = -A^T, A \in \mathbb R^n$, matricies of the form $R = \exp(A)$ are generalizations of rotations to higher dimensions called "special orthogonal matricies". If you want to rotate in the $e_j, e_{j+1} \ $ plane, you can take $i =\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} $ and use as your rotation matrix: $$R(\theta) = \exp(( 0_{j-1} \ \oplus i \theta \oplus 0_{n-j+1} \ \ \ )) = I_{j-1} \oplus \exp(i \theta) \oplus I_{n-j+1}$$
$0_k$ here is the k-by-k zero matrix and $I_k$ is the k-by-k identity matrix.
Note that we can define the complex numbers by $1 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ and $i = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$ so what's happening here is really the same algebra you already know and love. In higher-dimensions we can't just use one number, $\theta$ to describe all rotations, since the rotation can be happening in any plane. The only skew symmetric 2d matricies are those given by $i\theta$.
If you want rotate around a specific point instead of the origin we work inside homogenous coordinates of $\mathbb R^{n+1} \ $ where our points $p \in \mathbb R^n$ are represented by vectors $(p,1) \in \mathbb R^{n+1}$. A translation matrix just has the vector $(p,1)$ in the last column and 1's along the diagonal, just like how we represent 2-dimensional translations with 3x3 matricies. If $T_p$ is our translation matrix, we can write $T_p \ R(\theta) \ T_p^{-1}$ as our rotation in the $e_j, e_{j+1} \ $ plane around the point $p$.
If we call the column vector given by $(p,1)$ as $\mathbf p$, then we can write all rigid transformations as $\exp(A) \oplus \mathbf p$ with skew-symmetric $A$ which matches up with your $(a + bi) + \exp(i\theta)$.