Would it make sense to amp Klipsch S4s?
I'll be running the S4's from an iPod.
What amp would complement the S4s?
I probably wouldn't bother amping the S4, and if I did, I definitely wouldn't spend a lot of money. If you felt compelled to get an amp, I imagine you'd want something relatively portable to go along with the iPod and IEMs. Something like the Fiio E5 might be suitable for some additional power and headroom.
I certainly wouldn't amp the S4.
I can't say enough that I would only buy an amp if I had a headphone that my source couldn't drive properly. Since you haven't even bought the Shure SRH840 yet, why bother? The Shure is a portable headphone and IMHO has a high probabilty to run on your ipod just fine.
Just my 2 c.