Using the isomorphism ${\mathrm{Cl}}_{1,3}(\mathbb{C})\cong {\mathrm{Cl}}_{2,3}(\mathbb{R})$, it is possible to complexify the spacetime Clifford algebra ${\mathrm{Cl}}_{1,3}(\mathbb{R})$ by adding an additional timelike dimension to the Minkowski spacetime ${\mathbb{R}}^{1,3}$. In a recent work we showed that this treatment provide a particular interpretation of Dirac particles and antiparticles in terms of the two timelike coordinates. Throughout this article we study in detail the structure of the real Clifford algebra ${\mathrm{Cl}}_{2,3}(\mathbb{R})$, focusing on the isomorphism ${\mathrm{Cl}}_{1,3}(\mathbb{C})\cong {\mathrm{Cl}}_{2,3}(\mathbb{R})$ and on how to perform the embedding ${\mathrm{Cl}}_{1,3}(\mathbb{R})\hookrightarrow {\mathrm{Cl}}_{2,3}(\mathbb{R})$. On the first part of this paper we analize the Pin and Spin groups and construct an injective mapping $\mathrm{Pin}(1,3)\hookrightarrow \mathrm{Spin}(2,3)$. In particular we obtain elements in $\mathrm{Spin}(2,3)$ that represent parity and (unitary) time reversal in the Minkowski spacetime. On the second part of the article we study the space of algebraic spinors of the algebra and prove that the hermitian inner product on complex spinors in ${\mathrm{Cl}}_{1,3}(\mathbb{C})$ is reproduced in ${\mathrm{Cl}}_{2,3}(\mathbb{R})$ by the Clifford-conjugation inner product on real spinors.