The generalization of the Killing spinor equation to nonconstant Killing function $\lambda$ has been worked out in 

H.-B. Rademacher, <A HREF="http://link.springer.com/chapter/10.1007/BFb0083642#page-1">Generalized Killing spinors with imaginary Killing function and conformal Killing fields</A>, Lecture Notes in Math. 1481 (Springer, Berlin, 1991).

A nonconstant $\lambda$ is only possible if its real part is identically zero.

A more recent paper along these lines is <A HREF="http://arxiv.org/abs/1311.0969">Complex Generalized Killing Spinors on Riemannian Spin manifolds</A> (2013).