I use Audio-Technica ATH-M40X Professional Monitor Headphones for all my practicing and recording. I chose these over the more expensive M50X because they actually have a slightly flatter response.
They don’t sound “like a bass amp” - they are totally flat. I run my basses through a Zoom B1Four that allows me to model a variety of amp/cab combinations, as well as boost bass, scoop mids or use a gazillion other effects.
The ATH-M40X headphones replaced a pair of OneOdio Studio Pro 50’s which I now use for casual listening. They aren’t quite as clean as the Audio-Technicas, but still sound pretty good, especially considering that they cost half as much.