Traditional robust adaptive filtering algorithms are commonly designed by incorporating error nonlinearity cost functions and stochastic gradient descent (SGD) method to mitigate the effects of non-Gaussian/impulsive noises that can distort systems. However, these methods often face significant challenges when the input signal itself contains sharp spikes. This paper addresses the challenge of managing spikes and outliers often encountered in both input and noise signals within adaptive filtering settings. We propose utilizing the α-stable distribution model for both input and noise signals. Our approach centres on optimizing the modified Blake-Zisserman-based correntropy (MBZC) cost function and employing the fractional-order SGD (FoSGD) method, leading to the development of a new method called fractional-order MBZC (FoMBZC), that offers increased flexibility and broader applicability across various signal and noise types. We provide a detailed analysis of the mean stability and mean square steady-state behaviour of the FoMBZC under Gaussian conditions. Finally, simulation experiments in various non-Gaussian environments are performed to highlight the superiority of the proposed method compared to existing algorithms.