Antiferromagnetic spin fluctuations were investigated in the normal states of the parent ($x = 0$), under-doped ($x = 0.04$) and optimally-doped ($x = 0.06$) Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ single crystals using inelastic neutron scattering technique. For all the doping levels, quasi-two-dimensional antiferromagnetic fluctuations were observed as a broad peak localized at ${it Q} = (1/2, 1/2, l)$. At lower energies, the peak shows an apparent anisotropy in the $hk0$ plane; longitudinal peak widths are considerably smaller than transverse widths. The anisotropy is larger for the higher doping level. These results are consistent with the random phase approximation (RPA) calculations taking account of the orbital character of the electronic bands, confirming that the anisotropic nature of the spin fluctuations in the normal states is mostly dominated by the nesting of Fermi surfaces. On the other hand, the quasi-two-dimensional spin correlations grow much rapidly for decreasing temperature in the $x = 0$ parent compound, compared to that expected for nearly antiferromagnetic metals. This may be another sign of the unconventional nature of the antiferromagnetic transition in BaFe$_2$As$_2$.
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