Pretty much all* browsers will support 4096-bit keys. The issue you’ll run into is that key exchange is slower with larger keys, which will increase load on the server and slow down page loading on the client.
2048-bit keys are generally considered safe for the time being. If you want an intermediate step, though, 3072-bit keys are right smack-dab in the middle.
*: Only exception might be a couple of weird, old mobile / embedded browsers.