Here is an idea I've been working on for self study.
Suppose $S$ is a division subring of $\mathbb{H}$ (the quaternions, viewed as a subring of $M_2(\mathbb{C})$), which is stabilized by the maps $x\mapsto dxd^{-1}$ for all $d\neq 0$ in $\mathbb{H}$. Then either $S=\mathbb{H}$, or $S$ is contained in the center.
I suppose that $S\neq\mathbb{H}$. I define $\varphi_d\colon S\to S$ to be the conjugation map $x\mapsto dxd^{-1}$. Evidently, I find that each $\varphi_d$ is a bijection on $S$, as it is injective, and for any $y\in S$, $d^{-1}yd\in S$ and is the desired preimage for $\varphi_d$.
I also proved that $Z(\mathbb{H})=\mathbb{R}$. I don't know how to show $S\subset Z(\mathbb{H})$ when $S\neq\mathbb{H}$. I want to show $\varphi_d=\mathrm{id}_S$ for all $d\neq 0$ in $\mathbb{H}$, but I don't know how to prove that. Does anyone have any hints or suggestions on what to do?
I assume that $S$ contains at least one imaginary element $ai+bj+ck$. Conjugating by $i$ shows $ai-bj-ck\in S$, so $2ai\in S$. Similarly, $2bj,2ck\in S$. Is it possible to scale these to any real coefficient to conclude that $S$ contains all imaginary elements?