Identify $\mathbb R^4$ with the quaternionic field $\mathbb H$

Question

I would like to identify $\mathbb R^4$ with the quaternionic field $\mathbb H$. Then, this identification is given by : $$ (*) \quad \mathbb H \ni q= a +i b + j c + k d \longleftrightarrow (z,w) \in \mathbb C^2;\, z=a+i b, w=c+i d,$$ or by $$ (**) \quad \mathbb H \ni q= a +i b + j c + k d \longleftrightarrow (z,w) \in \mathbb C^2; \, z=a+i c, w=b+i d,$$ or other identification ?

Thank you in advance

Answer 1

(*) is closer to what tends to be useful. The reason is that $q=z+wj$ with this setup. Alternatively one can set

$$ (*) \quad \mathbb H \ni q= a +i b + j c + k d \longleftrightarrow (z,w) \in \mathbb C^2;\, z=a+i b, w=c-i d,$$

due to the fact that $$q=z+jw$$ in this case.