Anyone know what this is, and should I exterminate it?
Looks to be some sort of Caterpillar? But hanging by
webbing.

Yes Do-Not destroy it. That's a chrysalis I'm guessing some kind of caterpillar. Let's ask @mellielong she will probably know for sure.

First off, thanks for asking before exterminating. That is definitely a butterfly chrysalis. Looks like a Variegated Fritillary to me, but could be one of the other Fritillaries. Do you grow Passion Vine or Violets nearby? That's what the caterpillars eat.

Thank you Christine, yes it is a Chrysalis... says it a Metallic Mechanitis butterfly Chrysalis.... from Costa Rico

Sure is different looking...

Ooops Sorry Melanie,

Yes your probably right. I just looked up Chrysalis... matched pictures...
looked like Metallic Mechanitis, But it does also looks like Variegated
Fritillary, and I have lots of those caterpillars around this year.

Don't have Passion Vine, but do have loads of wild violets...

Thanks....

Glad to help!