Functional connectivity patterns in the human brain, like the friction ridges of a fingerprint, can uniquely identify individuals. Does this "brain fingerprint" remain distinct even during Alzheimer's disease (AD)? Using fMRI data from healthy and pathologically ageing subjects, we find that individual functional connectivity profiles remain unique and highly heterogeneous during mild cognitive impairment and AD. However, the patterns that make individuals identifiable change with disease progression, revealing a reconfiguration of the brain fingerprint. Notably, connectivity shifts towards functional system connections in AD and lower-order cognitive functions in early disease stages. These findings emphasize the importance of focusing on individual variability rather than group differences in AD studies. Individual functional connectomes could be instrumental in creating personalized models of AD progression, predicting disease course, and optimizing treatments, paving the way for personalized medicine in AD management.