It could be a Bonfire Nectarine which is used as grafting stock.

Scroll down about a third of the page of the link I provided above which states the following and shows a photo :

These are shoots from the rootstock. As is typical for fruit trees,
'Bonfire' is propagated by grafting. One popular rootstock is 'Royal Red
Leaf', which is what was used here. Peach trees usually have green
leaves, not purple, so with a purple-leaved rootstock it's immediately
clear when there are rootstock sprouts that need to be pruned away. With
'Bonfire', however, you need to identity these shoots by their point of
origin and their looser habit.

I did check out the entirety of the page you liked to, it had some useful information. We're not sure if we are just going to prune off the shoot or try to transplant it as it does appear to be rather beautiful all on its own. Thanks for the information.

You need to remove it one way or the other for your peach tree to bear properly. Otherwise the root stock can overwhelm the fruit tree.

the root stock is chosen.. because it is more vigorous than what was grafted onto it ... the converse is.. if the root stock had good fruit.. they wouldnt have used it as root stock ...

as such.. it will eventually take over the grafted plant ... and you might lose what you paid money for ..

often.. since it has aggressive roots .. what comes off the tree is not the best fruit .. in other words.. if you bought a prime peach eating tree .. this one is probably a lesser peach... or even an ornamental peach .. non fruiting ... the above links should say such ...

you didnt tell us what kind of peach you bought ... what your investment was [i dont need numbers] .... and whether the purple on fills the reason you planted a peach ... an ornamental peach.. purple.. that blooms in spring.. but never sets fruit.. can be a very nice ornamental plant for a garden ,.... if thats ok with you ...

btw.... i have given up on peach in MI ... they were a great experiment ... but they end up having so many problems.. and need spraying so many times a season ... i just quit ... it was cheaper and easier.. to just go buy a bushel at teh local orchard ... in season .. rather than invest all that time and money in spraying ... not to mention the exposure to such ...

anything you need to know.. about growing a fruit peach.. can be learned in the fruit forum ... just another base of experts .. that i usually dont see in the general forums ...

ken