It is an old theorem of Samelson that any compact Lie group $G$ of even rank has an integrable complex structure, which, in particular applies to the case of $\mathrm{SU}(3)$. Basically, one chooses a Cartan subalgebra $\frak{t}\subset\frak{g}$, which gives a splitting of the Lie algebra into (complex) root spaces, choose a base for the root system and then consider the positive root spaces, use those to define the $(1,0)$-vector fields and then define the complex structure however one likes on the Cartan sub-algebra $\frak{t}$ (which one can do because it has even dimension, by assumption), and then one shows that the left-invariant almost complex structure that this defines on $G$ is always integrable. (It follows almost immediately from the structure equations.)
Here is an explicit construction in the case of $\mathrm{SU}(3)$: Let $g:\mathrm{SU}(3)\to \mathrm{GL}(3,\mathbb{C})\subset M_{3,3}(\mathbb{C})$ be the standard inclusion homomorphism of $\mathrm{SU}(3)$ into the group of invertible $3$-by-$3$ complex matrices. Then $g$ satisfies $g^*g = I_3$ and $\det(g)=1$ (where $g^*$ means the conjugate transpose of $g$), so the canonical left invariant form $\gamma = g^{-1}\,\mathrm{d}g$ satisfies $\gamma^*+\gamma=0$ and $\mathrm{tr}(\gamma)=0$, so it can be written in the form
$$
\gamma = \begin{pmatrix}
i\theta_1 & \omega_3 & \omega_2\\
-\overline{\omega_3} & i\theta_2 & \omega_1\\
-\overline{\omega_2} & -\overline{\omega_1} & i\theta_3\end{pmatrix},
$$
where the $\omega_i$ are $\mathbb{C}$-valued $1$-forms and the $\theta_i$ are $\mathbb{R}$-valued $1$-forms satisfying $\theta_1+\theta_2+\theta_3 = 0$. By the definition of $\gamma$, we have the structure equation $\mathrm{d}\gamma = - \gamma\wedge\gamma$.
Now let $\lambda\in \mathbb{C}\setminus\mathbb{R}$ be any constant that has nonzero imaginary part and set $\omega_0 = \theta_1 + \lambda\,\theta_2$. Then the $\mathbb{C}^4$-valued $1$-form $\omega = (\omega_0,\omega_1,\omega_2,\omega_3)$ defines a left-invariant $\mathbb{C}^4$-valued coframing on $\mathrm{SU}(3)$, i.e., it provides a linear identification of each tangent space of $\mathrm{SU}(3)$ with $\mathbb{C}^4$, and hence defines a unique almost complex structure $J_\lambda$ on $\mathrm{SU}(3)$ for which the $1$-forms $\omega_i$ are of $J_\lambda$-type $(1,0)$. (This $J_\lambda$ really does depend on the choice of $\lambda$ above.)
Finally, the structure equation $\mathrm{d}\gamma = -\gamma\wedge\gamma$ implies
$$
\mathrm{d}\omega_i \equiv 0 \mod \omega_0,\omega_1,\omega_2,\omega_3
$$
for $0\le i\le 3$. (For example,
$$
\mathrm{d}\omega_1 = i(\theta_3-\theta_2)\wedge\omega_1 +\overline{\omega_3}\wedge\omega_2
\equiv 0 \mod \omega_0,\omega_1,\omega_2,\omega_3\,,
$$
the other cases are similar.) Thus, the Nijnhuis tensor of $J_\lambda$ vanishes, and so there is a unique complex structure on $\mathrm{SU}(3)$ for which the forms $\omega_i$ are of type $(1,0)$. (Because this almost complex structure $J_\lambda$ is left-invariant, it is a fortiori real-analytic, so the Newlander-Nirenberg theorem is not needed, the integrability follows from earlier results.)
