You could replace with a Refurbished iMac 27" computer.
The 27-inch models Apple released after your 2013 27" would be:
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Notes:
These are all Intel CPUs. Since there are as yet no Apple Silicon "M" series 27-inch iMacs, no one can say with certainty what Apple may release next year. So as with any technology purchase, you have to look at and weigh your needs, desires, requirements, $$, and etc. ;-)
The dates are the official Apple production/sale dates range.
The "Ci5" is short for Intel Core i5, the CPU Processor in the Mac, followed by the Intel code/common name of that CPU chip line.
The speed in GHz (GigaHertz) is the base/lowest frequency for that chip, all are able to boost into faster turbo mode when heavy demand is placed on them. And of course, Apple made higher CPU upgrade options available at time of order. The models I list are the base models.
OS listed is the range of Mac OS X (macOS) releases which that model can run.
There may be other specs which are also or more significant to you, as perhaps max RAM, size and kind of HDD/SSD storage or plug/ports available on the back of the iMac.
iMac (27-inch, Late 2013) <--{your model}
9/2013 to 10/2015
Ci5 4570 Haswell 3.2 GHz (4 cores)
OS X 10.8.4 Mountain Lion up to 10.15.7 Catalina (the last Mac OS for your model)
"Vintage" status
iMac (Retina 5K, 27-inch, Late 2014) -{no point really, very similar}
10/2014 to 10/2015
Ci5-4690 Haswell 3.5 GHz (4 cores)
OS X 10.10 Yosemite up to macOS 11 Big Sur
"Vintage" status
iMac (Retina 5K, 27-inch, Mid 2015) -{also, a little too similar}
5/2015 to 10/2015
Ci5-4590 Haswell 3.3 GHz (4 cores)
OS X 10.10 Yosemite up to macOS 11 Big Sur
"Vintage" status
iMac (Retina 5K, 27-inch, Late 2015) -{first later model with some staying power, but in "vintage" status}
10/2015 to 6/2017
Ci5-6500 Skylake 3.2 GHz (4 cores)
OS X 10.11 El Capitan up to macOS 12 Monterey
"Vintage" status
iMac (Retina 5K, 27-inch, 2017) -{first later model still in 'supported' status}
6/2017 to 3/2019
Ci5-7500 Kaby Lake 3.4 GHz (4 cores)
macOS 10.12 Sierra up to macOS 13 Ventura (highest macOS)
"Supported" status
iMac (Retina 5K, 27-inch, 2019) -{}
3/2019 to 8/2020
Ci5-8500 3.0 GHz Coffee Lake (6 cores)
macOS 10.14 Mojave up to macOS 13 Ventura (*current OS release)
"Supported" status
iMac (Retina 5K, 27-inch, 2020) -{the final 27" iMac (as of now...}
8/2020 to 3/2022
Ci5-10500 3.1 GHz Comet Lake (6 cores, hyper-threaded, 12 cpu activity threads)
macOS 10.15 Catalina up to macOS 13 Ventura (*current OS release)
"Supported" status
For help with deciding who is a reputable Refurbisher, see the MacRumors article:
Apple Refurbished Products: Should You Buy Them? Published May 9, 2022
https://www.macrumors.com/guide/apple-refurbished/
And the Refurbished Tracker site, RefurbMe
https://refurb-tracker.com/en/
Hardware Spec References:
EveryMac <
https://everymac.com/systems/apple/imac/index-imac.html>
MacTracker <
https://mactracker.ca/>