As an engineer, I learned a lot about how to use complex numbers. One way I have heard $i$, the unit complex number, defined is:
It is orthogonal to the real number line. Because $\frac{\mathrm{d}}{\mathrm{d}x} e^{x} = e^{x}$, we can clearly see $e^{\pi i} = -1.$
and the professor draws the unit circle.
However, this is not a unique definition of $i$. Whereas real 1 is $\hat{x}$ and $i$ represents $\hat{y}$ on the whiteboard, there are $\hat{z}$, $\hat{w}$, etc. which also satisfy the equations above.
We can use rotation to translate $\hat{z}$ to $\hat{y}$. Therefore it is still possible to use the "two-dimensional" definition $\mathbb{C}(a) <=> \hat{x} \Re(a) + \hat{y}(-\Im^2(a))$. In other words, if only care that $i^2 = -1$, then it does not matter which from the infinite set of $i$s you choose.
However we could consider $?(a) <=> \hat{x} \bullet a + \hat{y} \bullet a + \hat{z} \bullet a$ which satisfies $e^{\pi \hat{y}} = -\hat{x}$, $e^{\pi \hat{z}} = -\hat{x}$, and $e^{\pi \hat{y}\hat{z}} = -\hat{x}$.
Are these considered numbers in any way more useful than the normal complex numbers?
Note: I am not asking about $\mathbb{C}^3$, $\mathbb{C}^4$, ... as discussed at 3 Dimensional Complex Plane?
William Rowan Hamilton introduced quaternions in the 19th century. These generalize complex numbers and also generalize cross-products of vectors. A quaternion is an object of the form $$ a + bi + cj + dk \tag A $$ where $a,b,c,d$ are real numbers and $i,j,k$ are objects that can be multiplied as follows: \begin{align} i^2 = j^2 = k^2 & = -1 \\ ij = k & \qquad ji = -k \\ jk = i & \qquad kj = -i \\ ki = j & \qquad ik = -j \end{align} Addition and subtraction of quaternions is term by term.
Notice that $\pm i, \pm j, \pm k$ are not the only square roots of $-1$: if $a=0$ and $b^2+c^2+d^2=1$, then the square of the expression $(\mathbf A)$ above is $-1$.
The use of quaternions in physics was superseded by the use of vectors in $\mathbb R^3$ with the usual dot- and cross-products, but quaternions are used today in computer graphics.
Quaternions afford an easy way to see that the space of rotations of $\mathbb R^3$ that leave the origin fixed is not simply connected, as follows. First show that the map $$ bi+cj+dk \mapsto (Pi + Qj +Rk)\Big(bi+cj+dk\Big)(Pi + Qj +Rk)^{-1} $$ is a rotation of the $3$-dimensional space of "pure" quaternions ("pure" means the real part $a$ is $0$). Then observe that the two quaternions $\pm(Pi + Qj +Rk)$ both represent the same rotation. That means a path on the sphere from $Pi + Qj +Rk$ to $-(Pi + Qj +Rk)$ corresponds to a path in the space of rotations, from one particular rotation to itself, that cannot be contracted to a point.