Picked it up Saturday and today I have some problem or problems trip the check engine light. Scanned it with my Maxwell NT809 and get these codes.
U0121-00
U012A-00
U0294-00
B13C5-92
Is there some one item that would cause these? Bad PCM/BCM? Battery?
I appreciate any help figuring this out.

Need more info on the car. Year, mileage, age of battery,

First order of business would be to hook up a battery charger or maintainer and bring the battery up to a full charge. The car may have sat before you bought it. Even if it didn't you want to make sure the battery is fully charged. These cars are VERY finicky about batteries. Batteries with less than a full charge can lead to all kinds of problems that all magically disappear once the battery is fully charged.

More info and I'll get a charger hooked up as soon as I can.
2014 F Type base
Mileage a bit over 119K (Yeah, it's well used.)
PCM - U0447-00 Invalid data received from gateway A
P2183-22 Eng coolant temp sensor 2 circuit/performance
ATCM - U0121-00 Lost comm with ABS module
U012A-00 Lost comm with chassis control module
U0140-00 Lost comm with BCM
U0294-00 Lost comm with PCM monitor
Gateway - B13C5-92 Secondary battery
IPMB - U0140-08 Lost comm with BCM

Ok, I hooked up my smart charger. It tells me the battery is at 55% charge.
So, there's part of my problems. I hope fully charging clears the codes.
I also guess I need to replace it since it is 4 years old and should be at a higher charge with the driving I've been doing.

My battery is coming up to 5 years now and, at the risk of jinxing it, is still performing perfectly. I give it an occasional top-up with my CTEK overnight. I'm not convinced the charging system is the car can ever fully charge a battery unless it was pretty full to start with - I've read that it takes a 20 minute drive to just put back what was used for starting!

The 2014s use a second battery for start/stop. Once that battery dies, start/stop is no longer available. I eventually removed my 2nd battery after it was dead for a few years. I’m not aware that mine threw codes when it was dead, however.

It stores a code (or at least my 2015 does).

B13C5-92 SECONDARY BATTERY.
Performance Or Incorrect Operation.

Ah, that’s a code the OP reported. So he can ignore that one - it means that start/stop won’t work, but most people hate that feature. That shouldn’t trigger the CEL, it never did on mine.

Yes, should have been more clear, no CEL for me either!

Alright, battery changed, codes cleared. Car started and ran for a few minutes. No codes have reappeared.
I know I need to drive it for a while to make sure. But this is promising.
One correction to be made though. The scanner is a Foxwell NT809 not a Maxwell. Sorry about that Foxwell.

Excellent news. Thanks for updating us.

Scanned the car again and I have codes that returned.
Time to figure out these stored codes.
PCM
U0447-00 Intermittent - Invalid data received from gateway A
P2183-22 Temporary - Engine coolant temperature sensor 2 - circuit range/performance
All terrian control module
U0121-00 Intermittent - Lost communication with ABS control module
U012A-00 Intermittent - Lost communication with chassis module A
U0294-00 Intermittent - Lost communicaiton with PCM module
HVAC
B13C2-86 Intermittent - Windshield misting sensor
B13C2-96 Intermittent - Windshield misting sensor
Image processing module B
U0140-08 Intermittent - Lost communication with BCM

The coolant sensor one will most likely go away if I replace it. It is the rest that I need help diagnosing the probable causes.

Could these problems be caused by a flakey gateway module?

Gateway module needs to be reset, when you do a battery install you need the correct tools to do it ,like SDD or icarsoft v3 ,and theres a few other good ones ,what will happen your alternator wont charge your battery, so what you can do is take it to the dealer or any shop that has SDD and they will update your modules (GWM-BCM) AND BATTERY INSTALL PROCEDURE ,,YOU JUST NEED THE BATTERY PROCEDURE RESET CORRECTLY ,SO ALL MODULES TALK TO EACH OTHER,

Ok, thanks. I will see when I can get that done.