The noncommutative differential geometry of quantum flag manifolds has seen rapid growth in recent years, following the remarkable finding of a complex structure for flag manifolds of irreducible type by Heckenberger and Kolb. With a large part of the theory for the irreducible cases already figured out, it is now time to tackle the question of how to obtain the same structure for other types of flag manifolds. In this work in collaboration with R. Ó Buachalla and J. Razzaq, we give a complex structure for the full flag manifold of quantum SU(3) that includes the differential calculus discovered by Ó Buachalla and Somberg as its holomorphic sub-complex. I shall review this construction that makes use of Lusztig quantum root vectors, while at the same time giving a general overview of the theory of noncommutative differential calculi for quantum homogeneous spaces.