We study the emission of individual quantum dots in CdTe/ZnTe samples doped with a low concentration of Cr. In addition to dots with a photoluminescence (PL) split by the exchange interaction with a magnetic Cr atom, we observe another type of dots with a complex PL structure composed of a minimum of six lines on the exciton and biexciton and three lines on the charged excitons. In these dots, the linear polarization dependence and the magnetic field dependence of the PL behave like three similar quantum dots emitting at slightly different energies. Cross-correlation intensity measurements show that these emission lines are not independent but exchange intensities in a time scale of a few hundred nanoseconds depending on the optical excitation power. We attribute this PL structure to charge fluctuations of a Cr atom located in the vicinity the CdTe dots in the ZnTe barrier. We present a model which confirms that the presence of a single charge fluctuating between -e (Cr$^{+}$), 0 (Cr$^{2+}$) and +e (Cr$^{3+}$) and located a few nm away from the dot explains the observation of three emission energies. We finally show that the interaction between the confined carriers and the nearby fluctuating localized charge can be modified by an applied static electric field which modulates the splitting of the emission lines.
