803D4 - what integrated we thinking is a better match? Hegel? McIntosh? Other?

You can throw a lot of money away trying to make a pair of B&W speakers sound right. Some people can listen to them all day long. If you are one of those people, that's fine. You just need to make sure before you spend a lot of money. If you have a dealer, or some friends that can loan you an amp for a couple of days, do it.

Over the years, I've owned several pairs of B&W's (most were 800 series), and some of the brands I thought went well with them are, Rowland, Plinius, Belles, Pass and ARC. Just keep in mind that this is just my subjective opinion. What I like, you may not. I suspect McIntosh would be a good match, but I don't have enough experience with them to give a recommendation. I know most of their products have tone controls. They can be useful.

@pritchettsclosets 

Get any McIntosh with autoformers, it'll tame that egregiously aggressive high end that B & Ws have!

Depends completely on what specific improvements you’re looking for and what sound characteristics are most important to you. 

I've owned and heard at several audio shows B&W speakers paired with Musical Fidelity integrateds - good combo.

Have you considered the Primaluna EVO 300 Floyd integrated amp. I’m using is with an earlier B&W 803 diamond speaker and it’s great. It has plenty of grunt that the speaker needs and a tube preamp section. It’s really a fantastic amp. 
 

Good luck in your search.