Question: One channel of my split audio is cutting out when the cables are plugged into two devices.

I have a computer audio output that I have [this splitter](https://rads.stackoverflow.com/amzn/click/com/B000067RC4) connected to.

From the splitter I connect a cable to a standard headset.

Then I connect the other cable from the splitter to [this bass amplifier](https://rads.stackoverflow.com/amzn/click/com/B00HWINLAE)'s Input 1/4" jack (with an adapter). This is when the issue occurs.

Before I plug into the amp, the headset has sound coming from both ears. After I plug in, all sound stops coming through the right ear of the headset. Unplugging from the amp restores the sound to the right ear.

I can't duplicate the output into two separate jacks, so it has to be split from a single one somehow.  

The behavior is also not limited to that single headset, splitter, or cables. I've tried switching out every part of the setup except for the bass (since I only have one.)

I'm sure this is some normal phenomenon, but I can't find anything about it to save my life.

Answer: 
Guitar amp input impedance ≅ 1MΩ  

Headset impedance ≅ 50 - 300Ω  

Totally mis-matched.

However, this doesn't add up to what you're hearing. The amp should be the one suffering lack of signal, though it can make up the gain significantly.  

Best guess, as your amp is mono, it is shorting one side of the stereo signal, resulting in loss of one side on the headset.

In short [pardon the pun] - you really don't want to be doing it that way. Splitters like this really need to be splitting to two very similar devices. You'll need some kind of passive mixer at least to do this even vaguely properly.
Welcome to Sound Design.

Your bass amp is mono. The 1/4" input jack is mono. When you plug in to the 1/4" jack it shorts out your right channel so you lose the sound in the right channel of your headset.

This amp also has a 3.5 mm (1/8") auxiliary jack which accepts a stereo connection. Use that instead of the 1/4" main input and your headset should retain the right channel. (Inside the amplifier the left and right channels will be combined to give a mono sound output. However this will be done in a way that doesn't short out the right channel.)