Metaphor, discussing one concept in terms of another, is abundant in politics and can shape how people understand important issues. We develop a computational approach to measure metaphorical language, focusing on immigration discourse on social media. Grounded in qualitative social science research, we identify seven concepts evoked in immigration discourse (e.g. "water" or "vermin"). We propose and evaluate a novel technique that leverages both word-level and document-level signals to measure metaphor with respect to these concepts. We then study the relationship between metaphor, political ideology, and user engagement in 400K US tweets about immigration. While conservatives tend to use dehumanizing metaphors more than liberals, this effect varies widely across concepts. Moreover, creature-related metaphor is associated with more retweets, especially for liberal authors. Our work highlights the potential for computational methods to complement qualitative approaches in understanding subtle and implicit language in political discourse.