In the book Representation of compact Lie Groups of Tammo tom Dieck, chapter II.6, it is explained that if $V$ is a complex vector space and $W$ a quaternionic module, we have the isomorphisms (where the index is for the restriction of scalars) $$(\mathbb{H}\otimes_{\mathbb{C}}V)_{\mathbb{C}}\cong V\oplus \overline{V}$$ and $$\mathbb{H}\otimes_{\mathbb{C}}W_{\mathbb{C}}\cong W\oplus W$$ The first ${\mathbb{C}}$-isomorphism is given as $$(z_1+j z_2)\otimes v\to z_1 v\oplus z_2 v$$ where $\mathbb{H}$ is considered as a right ${\mathbb{C}}$-module. I fail to see why it works.
Edit
Actually I finally understand why it works like that, and no other way.
I had to go back to the Wikipedia definition of a change of ring, or of the tensor product of a $\mathbb{C}$-module with a $(\mathbb{H}, \mathbb{C})$-bimodule, which is a $\mathbb{H}$-module.
It is necessary to consider such structure if we want any action coming from the quaternionic action of the group on $\mathbb{H}\otimes V$ to commute with its structure as a $\mathbb{C}$-module.
It means here that $\mathbb{H}$ is considered as a right $\mathbb{C}$-module and as a left $\mathbb{H}$-module over itself. Then $\mathbb{H}\otimes_{\mathbb{C}}V$ is actually a $\mathbb{H}$-module where the multiplication by a quaternion $q'$ of $q\otimes v$ is $(q'q)\otimes v$. So the isomorphism given is a $\mathbb{C}$-isomorphism if $z_2v$ is considered an element of $\overline{V}$.
So only the following question remains:
End of Edit
And what is the second $\mathbb{H}$-isomorphism of $\mathbb{H}$-modules?
Think I got it now, but unfortunately it was more by hand than by a sound reasoning like in the case of complexification and de-complexication, where we looked for the eigenvalue of the complex structure.
So the second isomorphism that works seems to be, if $\mathbb{H}$ is always considered as a $(\mathbb{H}, \mathbb{C})$-bimodule, for $q\in\mathbb{H}$ and $w\in W_{\mathbb{C}}$ $$q\otimes_{\mathbb{C}} w\to q w\oplus q i w$$ The reciprocal of this isomorphism $W\oplus W\to \mathbb{H}\otimes W_{\mathbb{C}}$ is, for $v, w\in W$ $$(v, w)\to\frac12(1\otimes_{\mathbb{C}} v-j\otimes_{\mathbb{C}} j v)+\frac{i}2(1\otimes_{\mathbb{C}} w+j\otimes_{\mathbb{C}} j w)$$ The isomorphism is clearly $\mathbb{H}$-linear.
If somebody can validate my answer and give me a more systematic way of reasoning and / or reference about it, I will give him the bounty and considered his answer as accepted.