The new shoots should be cut out.

I'd try to snap them so they don't regrow but they are from under

I wonder if it is a weed seed that has germinated. possibly one of the sowthistle/hawksbeard type weeds. why not gently scrape down the stem of the incomer and see where it goes to. that will do little disturbance to the rose. if it isn't attached to the rose you will find out at least. from what I can see they don't look to be rose based.

I wonder if it is a pheasant berry shrub seed which the birds have donated to you? The leaf looks different to the rose leaves. The stem to the left of that purple stem looks like the pheasant berry shrub.

The leaves are an 1.5 incheslong & .5 wide max going to a point....Bird donation is what I thought but the new purple straight bit is odd. The rose looks happy tho....

I would give this pot a good soak and then tip out the contents. The water will hold the rootball together. On the side where you can see the stranger emerge (the thing I think is pheasant berry) scrape away the soil gently until you find where the roots are. The purple stalk may be attached to the stranger. If so get it out at the same time as the rest. If not get rid of stranger number one. You can repot that on its own if you want to see it grow up a bit. Keep scraping gently until you see where the second stranger is going. If it is attached to the rose pull it off. Do not cut it as it will just get stronger. Repot the rose using the soil you scraped away or new stuff if you want. Water it in then set it aside to recover. I know many will say do not move plants at the stage this rose has reached but I noticed I am not the only member who does what I think is best for a plant regardless of time of year. I try to get strangers out of pots before they intrude on the life of the plant I chose to grow in the pot. I often put a layer of newspaper just below the eventual surface height and put more soil or a mulch on top of that. It means strangers or weeds are easier to spot and remove early on. If the purple stem is from the rose, bin it. If it is a second plant then pot it up on its own and see what happens. Do let us know how you get on.

I dont think its a sucker.....Himalayan Honeysuckle looks fierce.....a mystery

My last post has not registered so another senior moment. The pheasant berry Leycestria Formosa is a nice shrub in the right place. Mine grows 10' + and the blackbirds and pheasants love the berries but they drop the seed wherever so I just weed them out. Easy when they are still small. The roots should not prove too difficult to get out. The stems are hollow which is helpful in id'ing it. The purple stem could be from the same plant. Looking forward to finding out what it is.

Leycesteria formosa is a lovely shrub and it can be cut back to fit your space. The blackbirds and pheasants love the berries. The stems are hollow which might help you identify it. The birds are not too good gardeners and drop the seeds wherever in the garden so I weed out lots of seedlings every year. Mine grows to over 10 feet no bother at all.

Ok . i put the pot on the operating table & eased everything out wuth 2 small hand forks. I was a bit keen not to get a splinter from the rose so I gently moved the hand fork in between & prized the visitor apart. Not much entanglement. I replaced the lost soil with new potting compost & topped up with small grit dressing.
The visitor has lots of leaves & new purple shoot but that's it. No ID. Its 2 ft high now!!