I guess if it makes sense to you, ok, but "continuous growth in a circle" to explain the complex exponential sounds like cloud-gazing to me: the human mind making up patterns where there are none.
The only way it makes sense to me is by looking at the power series and seeing how the exponential power series is almost the same as the sine and cosine power series (alternatively and equivalently: all of the exponential, hyperbolic, and circular functions satisfy almost the same differential equation).
Anyway, my point is that Euler's identity generalizes to any number that encapsulates "sideways" in Euclidean geometry. Any quaternion "x" that is a unit of rotation will satisfy "e^pi*x = -1". You can't get that easily from the power series understanding.
(I first got this picture from the Feynman lecture on algebra, http://www.feynmanlectures.caltech.edu/I_22.html, which doesn't seem to ever explain it that way explicitly. It's funny how memory works.)
Of course there's also value in looking deeper into the foundations.
In general the number of real numbers needed to describe a rotation is the number of rotational degrees of freedom. This number is n(n - 1) / 2 in n dimensions, or C(n, 2), basically the number of orthogonal (2-)planes of rotation in R^n.
[0] https://en.wikipedia.org/wiki/Degrees_of_freedom_(mechanics)
Similarly, when a quaternion is defined as representing a translation, that is likewise 3 degrees of freedom in 4 numbers. The four numbers would be similar to a real cartesian translation with dX, dY, dZ, and a fourth number = sqrt( dX^2 + dY^2 + dZ^2 ).
It's like the 4th number is a checksum for the other 3. You are adding a fourth number to use a more elegant mathematical calculation, which automatically keeps the dependent data relationships intact, like magic.
How do you encode translations with quaternions? Don't you need dual quaternions for that?
There may also be a representation that also encodes distance from the origin, but I am probably confusing this in my memory with something else.
I think of it like this: a complex number represents a rotation in a plane. You need two "coordinates" to do that. 2d has one plane, but 3d has two planes - think about how an anti-air cannon rotates left-right and up-down. So, if we've got two rotations to represent, you must need two complex numbers - that's four "coordinates", or one quaternion.
Just as in 2d with a complex, in 3d you can encode rotations with a 3d vector. The problem is that to reach any point from any other arbitrary point you can't use a single rotation, but a combination of two, sometimes three. Think about how your AAA gun operates or FPS viewpoint controls. Thus you start to use rotations along the three axes of a base which gives you a typical rotation matrix.
As soon as you start to compose rotations along axes you introduce a terrible phenomenon known as gimbal locking, where discontinuities appear especially near north and south poles.
And so, just as a 2D rotation in a plane can be represented by a vector orthogonal to that plane (i.e in a space outside the plane), the gimbal lock and all artifacts that are cumbersome when constraining ourselves in 3D can be dealt with by going up a dimension... Hence quaternions which encode+ a rotation by putting an arbitrary but well chosen axis and an amplitude in a space grown othrogonally to our 3D space.
+ Or isomorohically so
Why not 3 planes?
to be completely honest, I'm not sure if this intuition is correct. But it seems to make sense
Rotations in 2D are one-dimensional. Complex numbers are two-dimensional but we fix one degree of freedom by fixing the length.
Rotations in 3D are there-dimensional. Quaternions are four-dimensional but we fix the length again.
I then assumed that my idea with two rotations is how quaternions work. Here's to actually reading about a concept before I start explaining it next time...
And quaternions are made of two complex numbers.
edit: typo.
I don't pretend to grok quaternions (I only use them), but I'm fairly sure a quaternion consists of three imaginary numbers plus a real number.