Well, they look suspiciously like the tubers (or rhizomes, whatever botanically they might be) of my pink oxalis, O. tetraphylla.

If those new shoots turn into "4 leaved clovers" with a dark splosh in the center, that's what you've got.

Now there's a challenge! The "tubers" seem to be stacked vertically (new ones forming on top of old ones), which strikes me as being more like corms than tubers.

That makes me think monocot... arum is probably the right direction.

Perhaps it is a close relative of caladium?

It appears to be summer-dormant, and to be producing new shoots in fall, so how many arums do that? Arum Italicum is one that does, but I think the roots are much smaller.

Seems like it's already growing so I'd love to see how the leaves look in just a couple weeks when they mature.

Thanks everyone for looking at them. I will post back when they have grown. I potted them up and now just the wait- Love a mystery find! Should be quick I think. Hope it's something good.

I second the Oxalis proposal.

And, if it is - a pot is the safest place for them... :-(((

Here they are- these don't seem to be the same that I fight in the flower beds. They have small corms and little bulblets all around the bottom. I rip them out buy the handfuls. Scott Ogden in his book bulbs for the south seems to indicate that Oxalis crassipes has a thick starchy tuber with potato-like bublets on it. I've never seen that in my beds. Could this be it?
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Well, it's not O. tetraphylla, which has 4 leaflets per leaf, and is generally also variegated with a splash of purple-burgundy towards the center of the leaf.

Looks a lot more like Oxalis purpurea Grand Duchess Lavender to me. This one has 3 leaflets and the flowers look right to me. However, if the photo color is skewed on my screen or in your photos, it could be Grand Duchess Pink -- there really isn't that much difference in color between the two, the lavender is a pink-lavender, and the pink is a pepto bismol pink shade.

Do not let it loose in the garden...

But I say that about any/all Oxalis ;-))))

You will probably need to repot annually as they bulk up rapidly. Some are right at home in full sun and others prefer light shade/woodland. You look to have conditions 'just right' for that species.

Love that blue-green in the leaves. Very attractive.

I have Oxalis that makes little pink flowers and has corms that look just like those in your picture. I don't know why people don't like these. I sift through sod when I dig it up to make sure I don't toss any of these, which I did a lot of this year. I put them all around the "front" of my beds. Can't wait to see how they look next spring!