Can't find the eigenvectors of a simple 2x2 matrix

[This is what Wolfram tech support told me when I filed a bug report:]

That is because using machine precision (inexact) numbers such 0. (as opposed to exact numbers like 0) forces Eigenvectors[] to seach for eigenvectors numerically and hence the error message. For example, please evaluate and compare the results of the following expression used with machine precision and exact numbers:

Eigenvectors[{{Cos[1. x], Exp[I x]}, {1, 0}}]

Eigenvectors[{{Cos[1 x], Exp[I x]}, {1, 0}}]

This bug has been fixed as of Mathematica 10.3.

$Version

(* "10.3.0 for Mac OS X x86 (64-bit) (October 9, 2015)" *)

Eigenvectors[{{Cos[0. + x], Exp[I x]}, {1, 0}}]

(* {{-0.5 (-Cos[0. + x] + Sqrt[4 E^(I x) + Cos[0. + x]^2]), 1.}, 
    {-0.5 (-Cos[0. + x] - Sqrt[4 E^(I x) + Cos[0. + x]^2]), 1.}} *)

Simplify@Eigenvectors[{{Cos[0. + x], Exp[I x]}, {1, 0}}]

$\left( \begin{array}{cc} e^{-1. i x} \left(-0.25 \sqrt{1.+2. e^{2 i x}+16. e^{3 i x}+e^{4 i x}}+0.25 e^{2. i x}+0.25\right) & 1. \\ 0.25 e^{-1. i x} \left(\sqrt{1.+2. e^{2 i x}+16. e^{3 i x}+e^{4 i x}}+e^{2. i x}+1.\right) & 1. \end{array} \right)$