Denote $H$ as quaternion algebra and $C$ as complex number. Take any $a\in H$. One has $a=(a_1+ia_2)+(a_3+ia_4)j=c_1+jc_2$ with $ij=k$. Denote $M_n(F)$ as the matrix entries in division algebra $F$.
The book says $M_n(H)\to M_{2n}(C)$ is isomorphic. If this is true, then this must hold for $n=1$. So $H\cong M_{2}(C)$ as ring.
This homomorphism is given as the following. Using $c_1,c_2$ notation in first paragraph and labeling $x\in M_{2n}(C)$ as $x_{11},x_{12},x_{21},x_{22}$, I have $x_{11}=c_1,x_{12}=c_2,x_{21}=-\bar{c_2},x_{22}=\bar{c_1}$ where $\bar{x}$ denotes complex conjugation.
It is clear that this map is $R-$linear. So this had better to be isomorphic as $R-$vector space to start with even before talking about ring isomorphism.
Counting dimension over $R$ is $4\times 2=8$ for $M_{2}(C)$ and $H$ is only $4$. This cannot be isomorphism.
$\textbf{Q:}$ Is this a typo in the book? What is the correction? I could see it has to be identified to the image and this is clearly injection for sure.
Ref. D. Bump Lie Groups Chpt 5 Exercise 5.6
I am almost sure what is meant there is that it is an isomorphism to its image. Namely, the map is obviously not surjective. But you can check that it is an injective morphism of $\Bbb R$-algebras. Indeed, the case $n=1$ gives a standard representation of the quaternions as sub-$\Bbb R$-algebra of $M_2(\Bbb C)$.