We consider the difference Schrödinger equation ψ (z+h)+ ψ (z-h)+v(z)ψ (z) = 0, where z is a complex variable, h > 0 is a parameter, and v is an analytic function. As h → 0, analytic solutions to this equation have a simple WKB behavior near the points where v(z) ≠ ± 2. We study analytic solutions near the points z₀ satisfying v(z₀) = ± 2 and v′ (z₀) ≠ 0. These points play the same role as simple turning points for the differential equation ψ″ (z) + v(z)ψ (z) = 0. In an h-independent complex neighborhood of such a point, we derive uniform asymptotic expansions for analytic solutions to the difference equation.