I was reading Physics from Symmetry from Jakop Schwichtenberg and I got confused by the definitions of groups $U(1)$ and $SU(2)$.
As far as I understood, unit complex numbers with the ordinary multiplication forms a group and it is called $U(1)$. $U$ for its being unitary ($U^*U =1$) and $1$ for its being represented by single complex numbers.
Moreover, in the book he defines \begin{align} 1 =& \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} &i = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \end{align} and shows that we end up with the same results as $SO(2)$.
On the other hand, just like unit complex numbers, unit quaternions also form a group with ordinary multiplication. At this point he defines
\begin{align}\label{asd} &1 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} &i = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} &j = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix} &k = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix} \end{align} and called this $SU(2)$, S denotes $det(U)=1$ and $U$ denotes $U^\dagger U=1$ and 2 denotes $2\times 2$ matrices.
So, the question is: By the same logic we called unit complex numbers $U(1)$, shouldn't we say unit quaternions also $U(1)$, since they both unitary and represented by single number.
Also, By the same logic we called $SU(2)$ to matrix representation of the unit quaternions, shouldn't we also say that matrix representation of unit complex numbers are $SU(2)$
You are technically correct, but obviously this would cause a lot of confusion if you call the quaternion unitary group $U(1)$ as well. Note that if you say "represented by a single number", it matters if you mean real, complex or quaternion numbers. Therefore, the quaternion unitary group is usually denoted $U(1,\mathbb{H})$ or $Sp(1)$. As you correctly observed, it holds $SO(2) \cong U(1)$ and $SU(2) \cong U(1,\mathbb{H})$.