This study investigates the potential of fractal-based complexity measures as early warning indicators of stock market crashes, focusing on the S&P 500 index from 2000 to 2025. We adapt the relative roughness (RR) methodology, originally developed in physiological research, to financial market data and integrate it with the Hurst exponent (H) and Detrended Fluctuation Analysis (DFA) scaling exponent (α). Using a sliding window approach, we analyze raw prices and volatility series to capture time-varying patterns preceding major downturns. Our findings reveal consistent pre-crash anomalies: (i) for initial prices of S&P index H and α tend to decline during crisis phenomena, reflecting a loss of long-range correlations, while RR generally increases, indicating heightened short-term variability relative to long-term trends; (ii) for volatility series, during crashes, H and α rebound, while RR decreases sharply, signifying a shift toward strong persistence in volatility. These dynamics align with the Fractal Market Hypothesis (FMH) and the concept of critical slowing down in complex systems. The results suggest that RR, in combination with H and α, can serve as a computationally simple yet powerful diagnostic tool for market instability detection.