In a large saucepan, combine the pineapple with the onion, bell pepper, sugar, vinegar, cardamom and ginger and let stand at room temperature for 1 hour, stirring occasionally. Bring to a boil over high heat, stirring occasionally. Add the cranberries and cook over moderately low heat until the pineapple is translucent and the cranberries have popped, about 40 minutes. Season with salt; let cool. Transfer to jars and refrigerate.
Make Ahead: The chutney can be refrigerated for 1 month.