Here are some formula manipulations that yield complex-free results:
sol = DSolve[{y'[x] == y[x]/(x + Sqrt[x^2 + y[x]^2]),
y[0] == c}, y[x], x];
Simplify[sol, c > 1];
% // PowerExpand // TrigToExp;
Simplify[%, c > 1] // PowerExpand // ExpandAll // Simplify // Union
Check the results before you use them, because PowerExpand
uses assumptions that may not be true. These may or may not additional solutions:
Simplify[sol, -1 < c < 1];
% // PowerExpand // TrigToExp;
Simplify[%, -1 < c < 1] // PowerExpand // ExpandAll //
Simplify // Union
Simplify[sol, c < -1];
% // PowerExpand // TrigToExp;
Simplify[%, -1 < c < 1] // PowerExpand // ExpandAll //
Simplify // Union