In this paper, the author studies the quasiclassical asymptotics of analytic solutions to the equation $$ \Psi(z+h)=M(z)\Psi(z), \tag 1 $$ where $z$ is a complex variable, $h$ is a positive parameter, $M$ is a given $SL(2,\Bbb{C})$-valued function analytic in some domain $D$, and $\Psi$ is a vector solution. \par This problem (or its equivalent) occurs in many fields of mathematics and physics: in solid state physics, in wave diffraction and propagation theory, in the theory of almost periodic equations, in the theory of orthogonal polynomial, etc. \par For the differential equation $$ ih\frac{d\Psi(z)}{dz}=M(z)\Psi(z), \tag 2 $$ the parameter $h$ is also a quasiclassical parameter. Asymptotics for solutions of (2) are described by using a method that sometimes is referred to as the complex WKB method. This method has a huge number of applications. The study of analytic solutions to Equation (1) is more complicated because (1) is not local in $z$, and the space of its analytic solutions is infinite-dimensional. For the difference equations, there is an analog of the complex WKB method. In this paper, the author formulates basic theorems about the existence of analytic solutions to (1) having standard quasiclassical behavior, proves their existence in the case when $M$ is analytic in a bounded domain, and considers the case when $M$ is a trigonometric polynomial. A simple method to derive the asymptotics of solutions as $h\to 0$ is described.