  We start developing a formalism which allows to construct supersymmetric
theories systematically across space-time signatures. Our construction uses a
complex form of the supersymmetry algebra, which is obtained by doubling the
spinor representation. This allows one to partially disentangle the Lorentz and
R-symmetry group and generalizes symplectic Majorana spinors. For the case
where the spinor representation is complex-irreducible, the R-symmetry only
acts on an internal multiplicity space, and we show that the connected groups
which occur are SO(2), SO(1,1), SU(2) and SU(1,1). As an application we
construct the off-shell supersymmetry transformations and supersymmetric
Lagrangians for five-dimensional vector multiplets in arbitrary signature
(t,s). In this case the R-symmetry groups are SU(2) or SU(1, 1), depending on
whether the spinor representation carries a quaternionic or para-quaternionic
structure. In Euclidean signature the scalar and vector kinetic terms differ by
a relative sign, which is consistent with previous results in the literature
and shows that this sign flip is an inevitable consequence of the Euclidean
supersymmetry algebra.
