A four-dimensional real algebra 𝔄 is left ¢-associative (with rapect to ℭ) if 𝔄 is a bimodule with respect to a subalgebra ℭ isomorphic to the complex numbers, and c(AB) = (cA)B for all A, B in 𝔄 and all c in ℭ. We reduce the division algebra condition for such algebras to a question about a single-variable quartic, classify the division algebras, and show they all have nullity one. We determine the derivation algebras and find intersections with other classes of algebras. Right ¢-associative algebras are considered via the opposite algebra.