I'm assuming that you have a copy of Dominic Joyce's 1992 JDG article, "Compact hypercomplex and quaternionic manifolds" handy.  Write $\frak{g} = \frak{su}(3)$  as a direct sum
$$
\frak{su}(3) = \frak{b}\ \oplus\ \frak{d}\ \oplus\ \frak{f}\ ,
$$
where
$$
{\frak{b}} = \left\{\begin{pmatrix}ia&0&0\\0&ia&0\\0&0&-2ia\end{pmatrix}\ \Biggl|\ a\in\mathbb{R}\ \right\}\simeq \mathbb{R}
$$
$$
{\frak{d}} = \left\{\begin{pmatrix}ip&q+ir&0\\-q+ir&-ip&0\\0&0&0\end{pmatrix}\ \Biggl|\ p,q,r\in\mathbb{R}\ \right\}\simeq{\frak{su}}(2)
$$
$$
{\frak{f}} = \left\{\begin{pmatrix}0&0&-\overline z\\0&0&-\overline w\\z&w&0\end{pmatrix}\ \Biggl|\ z,w\in\mathbb{C}\ \right\}\simeq \mathbb{C}^2\simeq\mathbb{H}.
$$
One can easily check that this decomposition satisfies all of the conditions of Lemma 4.1 on page 751 (where, since $n=1$ in this case, I'm omitting the subscripts and summation signs).  Now, in Theorem 4.2, we have $k=m=0$, and we can define the complex structures $I_1$, $I_2$, $I_3$ on ${\frak{su}}(3)$ satisfying $I_1I_2=I_3$ as on pages 753–754.  (This depends on choosing a basis of $\frak{b}$, so there is a $1$-parameter family of such choices.)  By Lemma 4.3, these extend by left-invariance to integrable almost complex structures on $G=\mathrm{SU}(3)$, and so they define a left-invariant hypercomplex structure on $\mathrm{SU}(3)$.