Well it is always tricky trying to answer the question "why do things work they way they do" because the answer is usually a moving target.
Unfortunately, not one lectures or books what I saw or read don't explain principe of working of quaternion in 4d,
Well, Hamilton was not really 'thinking in $4$ dimensions' AFAIK... he was very interested in doing three dimensional geometry with quaternions. If you still think this is a sticking point you will have to explain what exactly you want to understand. As with most mathematics, you can get a get a long way by simply "getting used" to how things work, and then forming your own pictures as you go. Expecting a clear picture from the outset is often too ambitious.
I explain it for myself as three Cartesian plane with the one common real component which defines a vector position in 4d.
I do not think it is useful to think of quaternions as a vector position in $4$-D (that is going no further than thinking of $\mathbb R^4$.) Complex numbers are certainly a lot more than vector positions in $2$-D. You could think of $3$-space as two orthogonal Cartesian planes meeting on a real line, and you can think of $4$-space as three mutually orthogonal planes meeting on a real line. But this has more to do with $\mathbb R^3$ and $\mathbb R^4$ and not really anything to do with the quaternions.
This is my attempt to somehow imagine a quaternion.
Why is imagining quaternions any more challenging than imagining integers, rational numbers, or real numbers? It's just another, albeit different, number system that you can add, multiply and divide in. IMO more strenuous attempts to "imagine" ("imagine as an object in reality"?) do not yield anything useful compared to the amount of thinking that goes into it.
Is it far from reality?
I think we see this question sometimes, but it doesn't have an answer. I don't know what reality you're talking about. The usefulness of whatever picture a person has is relative to their own understanding of the subject, and stands on the merits of its own appeal. There is no standard of reality to measure a description against.
Why we disregard of real part of coordinate in first part of Euler formula?
I don't know what you're talking about. As I understand it, we do not disregard that part. I will give you my heuristic for rationalizing quaternion rotation below. I'm excerpting a couple slides from a talk I gave on quaternions:
Helpful identities
If the coefficients of $q$ have Euclidean length $1$, then $q^{-1}=\bar q$.
If $v$ and $w$ have real part zero, then
- The real part of $vw$ is $-1(v\bullet w)$.
- The pure quaternion part of $vw$ is $v\times w$.
If $u^2=-1$, $(\cos(\theta)+u\sin(\theta))(\cos(\theta)-u\sin(\theta))=\cos(\theta)^2+\sin(\theta)^2=1$ (basic trigonometry)
If $u^2=-1$, $(\cos(\theta)+u\sin(\theta))(\cos(\theta)+u\sin(\theta))=\cos(2\theta)+u\sin(2\theta)$ (De Moivre's formula)
Rationalizing quaternions' multiplication action on the model of $3$-space
The model of $3$-space I'm referring to, of course, is the space of quaternions with real part zero. As usual, we take $q = \cos(\theta/2)+u\sin(\theta/2)$ as the rotation quaternion, where $u$ is a unit vector pointing along the axis of rotation and $\theta$ is the angle of rotation around the axis measured using the right-hand rule. We aim to make the 'sandwich' action look more like what happens in complex arithmetic
$u$ is unmoved by $q$: $$quq^{-1}=
(\cos(\theta/2)+u\sin(\theta/2))u(\cos(\theta/2)-u\sin(\theta/2))=\\
(\cos(\theta/2)+u\sin(\theta/2))(\cos(\theta/2)-u\sin(\theta/2))u=\\
(\cos(\theta/2)^2-(u\sin(\theta/2)^2))u=\\
(\cos(\theta/2)^2+\sin(\theta/2)^2)u=u$$
if $v$ is a unit length pure quaternion orthogonal to $u$: $$qvq^{-1}=
(\cos(\theta/2)+u\sin(\theta/2))v(\cos(\theta/2)-u\sin(\theta/2))=\\
(\cos(\theta/2)+u\sin(\theta/2))(\cos(\theta/2)+u\sin(\theta/2))v=\\
(\cos(\theta/2)+u\sin(\theta/2))^2v=\\
(\cos(\theta)+u\sin(\theta))v\leftarrow\text{looks like a rotation in the complex plane}$$
$q$ leaves $u$ unchanged and rotates its normal plane by $\theta$. Everything else follows rigidly, so we have the rotation explained in terms that look like complex arithmetic.
There is one critical thing to notice here, though: in the last expression, $u$ and $v$ would both be $i$ in complex arithmetic. Let me try to explain that. The circle of quaternions that cause rotations around $u$ live in the plane $P$ spanned by $1$ and $u$. They are acting on the set $P^\perp$, the orthogonal complement of the first plane. You can see we need at least $4$ dimensions to fit these together in the same space.
I don't know the right way to explain how this can be aligned with complex multiplication, but I believe there is a concrete rationalization. In simple terms, I think it has to do with shifting perspective that the things you are operating on live in $P^\perp$ to operating on things in $P$. In harder terms, I believe it has to do with a duality between $P$ and $P^\perp$, which I've seen explained in some texts on Clifford/geometric algebra, but I do not properly know.
Final word
I believe also there is another good explanation of how the 'sandwich' action arises, an explanation that relies on exponential maps and Lie algebra, which again I have not fully absorbed. I leave it to someone who is more familiar with that field to provide a complementary answer along those lines.
