Complex representation of a quaternionic matrix It is evident that right module $\mathbb{H}^n$ is $\mathbb{C}$-linearly isomorphic to $\mathbb{C^{2n}}$ with corresponding isomorphism   $\nu :  \mathbb{C^{2n}} \to\mathbb{H}^n   $ given by $ \nu(a,b) = a + b\mathrm{j}$. This naturaly gives  representation for any quaternionic matrix $M \in \mathcal{M}^{n \times m}(\mathbb{H}) $ with two complex matrices $A,B \in \mathcal{M}^{n \times m}(\mathbb{C})$ as $M = A + B\mathrm{j}$.
It's assumed that  complex matrix representing $\nu^{-1}M\nu$ in parallel with complex representation of quaternion numbers can be written in  the form
$$
\theta_{n,m}(M) = \theta_{n,m}(A+B\mathrm{j}) =
 \left[\begin{matrix} A & B \\ -\overline B & \overline{A} \end{matrix}\right]$$
where $\overline{A}$ is a complex conjugate. However, what i don't understand is there this conjugation came from and I need your help.
When I write
$$ \nu^{-1}M\nu(a,b) = \nu^{-1}(A +B\mathrm{j})(a + b\mathrm j) =\nu^{-1}\left(Aa + Ab\mathrm{j} + B\overline{a}\mathrm{j} -B\overline b\right) = \left( Aa - B\overline{b}, Ab + B\overline a \right) $$
I don't have any idea what to do with conjugates to show that this map even linear.                
 A: Firstly, I want to thank Jyrki Lahtonen, Arctic Tern and mathreadler for elevating my understanding.
The key reason of my confusion was usage of left matrix on vector multiplication in  a right module $\mathbb{H}^n$. As it turns out in right modules it would safe only to use corresponding matrix ring  acting on the module from the right. Hence, if we assume that map $\nu$ also act from the right $(a,b)\nu = a + b\mathrm{j} $ we can  get desired result
$$ (a,b)\nu M \nu^{-1} = (a +b\mathrm{j})(A +B\mathrm{j})\nu^{-1} 
  = ( aA + aB\mathrm{j} +b\overline{A}\mathrm{j} -b\overline{B} )\nu^{-1} = (aA  - b\overline{B},aB + b\overline{A}) $$
which will yeld correct matrix representation if we assume that $aA \triangleq A^\top a^{(\top)}$.  
Otherwise we can just define $\mathbb{H}^n$ as a left module and get similar result.  
So we can actually think about $\theta_{n,m}$ as  change of global charts if we think about quaternionc matrix M as a (nonlinear) function $\mathbb{H}^n \to \mathbb{H}^m $.
