We consider divergence form elliptic equations Lu:=∇⋅(A∇u)=0 in the half space R+n+1:={(x,t)∈Rn×(0,∞)}, whose coefficient matrix A is complex elliptic, bounded and measurable. In addition, we suppose that A satisfies some additional regularity in the direction transverse to the boundary, namely that the discrepancy A(x,t)−A(x,0) satisfies a Carleson measure condition of Fefferman–Kenig–Pipher type, with small Carleson norm. Under these conditions, we establish a full range of boundedness results for double and single layer potentials in Lp, Hardy, Sobolev, BMO and Hölder spaces. Furthermore, we prove solvability of the Dirichlet problem for L, with data in Lp(Rn), BMO(Rn), and Cα(Rn), and solvability of the Neumann and Regularity problems, with data in the spaces Lp(Rn)/Hp(Rn) and L1p(Rn)/H1,p(Rn) respectively, with the appropriate restrictions on indices, assuming invertibility of layer potentials for the t-independent operator L0:=−∇⋅(A(⋅,0)∇).