Differential equations in the complex plane are different from differential equations on the real line.
Suppose you have an nth order linear differential equation as follows.
The real case
If the a‘s are continuous, real-valued functions on an open interval of the real line, then there are n linearly independent solutions over that interval. The coefficients do not need to be differentiable for the solutions to be differentiable.
The complex case
If the a‘s are continuous, complex-valued functions on an connected open of the real line, then there may not be n linearly independent solutions over that interval.
If the a‘s are all analytic, then there are indeed n independent solutions. But if any one of the a‘s is merely continuous and not analytic, there will be fewer than n independent solutions. There may be as many as n-1 solutions, or there may not be any solutions at all.
Source: A. K. Bose. Linear Differential Equations on the Complex Plane. The American Mathematical Monthly, Vol. 89, No. 4 (Apr., 1982), pp 244–246.