  Because of the isomorphism ${C \kern -0.1em \ell}_{1,3}(\Bbb{C})\cong{C \kern
-0.1em \ell}_{2,3}(\Bbb{R})$, it is possible to complexify the spacetime
Clifford algebra ${C \kern -0.1em \ell}_{1,3}(\Bbb{R})$ by adding one
additional timelike dimension to the Minkowski spacetime. In a recent work we
showed how this treatment provide a particular interpretation of Dirac
particles and antiparticles in terms of the new temporal dimension. In this
article we thoroughly study the structure of the real Clifford algebra ${C
\kern -0.1em \ell}_{2,3}(\Bbb{R})$ paying special attention to the isomorphism
${C \kern -0.1em \ell}_{1,3}(\Bbb{C})\cong{C \kern -0.1em \ell}_{2,3}(\Bbb{R})$
and the embedding ${C \kern -0.1em \ell}_{1,3}(\Bbb{R})\subseteq{C \kern -0.1em
\ell}_{2,3}(\Bbb{R})$. On the first half of this article we analyze the Pin and
Spin groups and construct an injective mapping
$\operatorname{Pin}(1,3)\hookrightarrow\operatorname{Spin}(2,3)$, obtaining in
particular elements in $\operatorname{Spin}(2,3)$ that represent parity and
time reversal. On the second half of this paper we study the spinor space of
the algebra and prove that the usual structure of complex spinors in ${C \kern
-0.1em \ell}_{1,3}(\Bbb{C})$ is reproduced by the Clifford conjugation inner
product for real spinors in ${C \kern -0.1em \ell}_{2,3}(\Bbb{R})$.
