I have a little problem wrapping my head around quaternions, in particular I have problems about how to pair the usual "3D algebra" with the theoretic vision of a quaternion.

I know that informally a quaternion is a structure composed of 3 imaginary parts and 1 coefficient/real part .

Now back to elementary number theory, I know that I can visualize a pair composed of an imaginary number + a real part in $\mathbb{R^2}$, the so called complex numbers can be visulized this way with all their properties.

The problem comes when I have 3 imaginary parts and only 1 real, even more, I have to somehow group this into the same family with Euler angles and cartesian coordinates .

I know that mathematicians usually skip this part quite easily by telling you that $\mathbb{i}$ is just a "device" but I know a have a quaternion that is effectively using 4 dimensions, working in a 3 dimensional space while being comparable to other " native $\mathbb{R^3}$ solutions " such as Euler angles .

My question is why quaternions works so well in a 3D metric space and is a quaternion a 4D object ?