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.