Significance The El Niño Southern Oscillation (ENSO) has significant impact on global climate and seasonal prediction. A simple modeling framework is developed here that automatically captures the statistical diversity of ENSO. In addition to simulating different types of El Niño and La Niña with realistic features, the model succeeds in capturing both the variance and the non-Gaussian statistical properties in different Niño regions spanning the Pacific. Particularly, the observed episode during the 1990s, where a 5-y central Pacific El Niño is followed by a super El Niño and then a La Niña, is reproduced by the model. Key features of the model are state-dependent stochastic wind bursts and nonlinear advection of sea-surface temperature that allow effective transitions between different ENSO states.