I think you successfully answered your own question, but I'll elaborate a bit.
Usually, the Lorentz Group refers to SO(3,1). The fundamental representation of these group elements is by 4 x 4 matrices with real numbers in them. Since all higher dimensional matrix representations can be built using an outer product of fundamental reps, all the higher dimensional rep matrices only have real numbers in them. These matrices will only cause linear combinations with real coefficients of the basis vectors in the reps carrier space...therefore I guess you could call it a real vector space.
These SO(3,1) irreps are able to rotate/boost many (but not all) objects found in the physical world. These irreps are only good for integer spin objects. None of these irreps can rotate half integer spin objects (spinors)(like electrons).
However,there is another group SL(2,C) for which there are 3 complex Lie group parameters that specify each group element. For SO(3,1) there are 6 real Lie group parameters (3 rotation angles, 3 boost parameters) that specify each group element. You can stuff the 6 real parameters into the 3 complex parameters using the imaginary $i$. SL(2,C) is called the "covering group" of SO(3,1). The generators of the two groups can be put in a correspondence with each other such that both sets of generators have the same commutation relations. The group elements of the fundamental rep of SL(2,C) are 2 x 2 matrices with complex numbers in them. All the higher dimensional irreps are built of outer products of these complex 2 x 2 matrices. Some of these higher dim matrices still have complex numbers. Others end up with just real numbers and duplicate the real irreps of O(3,1). So, many of the SL(2,C) irreps will cause linear combinations with complex coefficients of the basis vectors in the irreps carrier space...therefore I guess you could call it a complex vector space.
The SL(2,C) irreps are able to rotate/boosts all objects found in the physical world (ie: both the half integer and integer spin particles).
The pedantically correct opening lines of your question should be:
In Michele Maggiore's book "A Modern Introduction to QFT" he describes the spinorial representations of SL(2,C), the covering group of the Lorentz group as
The representations are in general complex.