  We show how the Bell correlations can be modelled locally by relaxing the
joint probability relation for independent variables $P(a,b)=P(a)P(b)$ outside
classical settings, with complex/quaternion generators for the measurement
outcomes encoding a deviation from this (classical) requirement. The
consideration is motivated by complementarity. We analyse the pair correlations
(photons or spin 1/2 particles) based on the limitations put on complementary
observables when they are modelled as classical, non-commuting operators. As
complementarity comes with both single particle properties and correlations
insufficiently captured by classical theory together with `non-commutativity',
we argue that a complete model should allow complementary observables to have
new local dependences. That is, the existing information should not be
restricted to be classical (simultaneously observable). Given this new
correlation model, the central question is if information theory ought to be
extended accordingly in quantum physics.
