  We construct a quantization of the moduli space
$\mathcal{GH}_\Lambda(S\times\mathbb{R})$ of maximal globally hyperbolic
Lorentzian metrics on $S\times \mathbb{R}$ with constant sectional curvature
$\Lambda$, for a punctured surface $S$. Although this moduli space is known to
be symplectomorphic to the cotangent bundle of the Teichm\"uller space of $S$
independently of the value of $\Lambda$, we define geometrically natural
classes of observables leading to $\Lambda$-dependent quantizations. Using
special coordinate systems, we first view
$\mathcal{GH}_\Lambda(S\times\mathbb{R})$ as the set of points of a cluster
$\mathscr{X}$-variety valued in the ring of generalized complex numbers
$\mathbb{R}_\Lambda = \mathbb{R}[\ell]/(\ell^2+\Lambda)$. We then develop an
$\mathbb{R}_\Lambda$-version of the quantum theory for cluster
$\mathscr{X}$-varieties by establishing $\mathbb{R}_\Lambda$-versions of the
quantum dilogarithm function. As a consequence, we obtain three families of
projective unitary representations of the mapping class group of $S$. For
$\Lambda <0$ these representations recover those of Fock and Goncharov, while
for $\Lambda\geq 0$ the representations are new.
