Why complexify in order to construct Dirac representation? Suppose we have a theory is covariant under the Spin group Spin(2n-1; 1). We consider the real vector space $V = R^{2n-1,1}$, which naturally comes with a Lorentzian inner product. On this vector space we introduce an orthonormal basis $e_0; e_1; ... ; e_{2n-1}$, where $e _0$ denotes the time direction.
To construct the Dirac representation of Spin(2n-1; 1) we take the complexified space $T = \mathbb{C} <e_1; ... ; e_n>$. My question is why is it that in order to construct the Dirac representation we complexify the space?
NOTE: Theory is even dimensional and of Lorentzian signature.
 A: We assume that OP asks apart from the facts that: 


*

*Dirac representations by definition are complex;

*It is much easier to work with an algebraically closed field;

*Any real representation can be extended to a (possibly reducible) complex representation, so one is not missing anything by going complex.
In other words, OP is interested in why certain real Lie group representations cannot exist. Since it is well-known that every Lie group representation induces a corresponding Lie algebra representation, it will be enough for our purpose to show that certain real Lie algebra representations cannot exist.
So we are interested in whether there exists an $2^{[\frac{n}{2}]}$-dimensional$^1$ real spinor representation of $so(p,q)$, where $n=p+q\geq 2$?
A low dimension where this fails is $(p,q)=(3,0)$, i.e. 3D rotations, where we leave it as an exercise for the reader to check that the 1-dimensional pseudoreal/quaternionic spinor representation of the Lie algebra $so(3)\cong su(2)\cong u(1,\mathbb{H})$ has no real 2-dimensional irreducible subrepresentation.
OP only asks about even dimension $n$ with Minkowski signature. One may similarly show that $(p,q)=(5,1)$ fails, i.e that the direct sum of the 2-dimensional left and the 2-dimensional right pseudoreal/quaternionic Weyl spinor representations of the Lie algebra $so(5,1)\cong sl(2,\mathbb{H})$ has no real 8-dimensional irreducible subrepresentation.
Incidentally, Witten recently discussed real, pseudoreal and complex representations of fermions in arXiv:1508.04715.  
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$^1$ To understand where the dimension $2^{[\frac{n}{2}]}$ comes from, see e.g. this Phys.SE post.  
