Strong and explosive athletes have a tendency to get locked in the sagittal plane (driven into extension). If this becomes excessive, they lose access to their frontal and transverse planes, which can cause injuries up and down the kinetic chain. To remedy this, offset push-up variations can be crucial in getting trunk rotation back.
Push-ups in this category can include offset variations off a box (see video) and offset variations with one hand on a medicine ball.
Based on your needs and goals there are tons of different options at your disposal. And if you want the best of all worlds, simply rotate your emphasis every 2-3 months to help build a strong, well-balanced, and bulletproof physique.