Q: How do I set an alarm on my watch without it also being set on my iPhone?

So you are setting an alarm on the watch itself, and it is appearing in the alarm list on the phone? Or is this just alerting you on the phone?

According to this page of the Apple Watch User Guide, http://help.apple.com/watch/#/apd27ce65478  it would work the other way around, but an alarm set on the watch should not be pushed to the iPhone or alert on the iPhone. I just tested this on my watch while writing this response and an alarm on my watch did not appear on the phone or alert on the phone. Is it possible that you did not create a new alarm, but rather edited an existing one?  Sorry, that doesn't matter, as alarms set on the phone do not actually appear on the watch, only the alert is pushed to the watch.

Thanks for the update. I've not experimented with the bedtime feature yet. Something I will start over the Christmas break. I'm a college professor, and during the semester, my sleep schedule is no where near a schedule!

Jim Constant wrote:

 

I figured it out, Chris. Everything you said was correct so this morning I started poking around in other places and discovered that it's the new Bedtime feature. I noticed that my phone had the little alarm clock icon in the status bar but I had no alarms set in the alarm tab of the Clock app. That was a clue to me to look at the other tabs of the Clock app where I saw what was happening. I had the watch alarm set to go off at 5am and told the Bedtime feature that I get up at 5am so they were both, independently, telling me it was 5am.

This is only tangentially related to your problem but I thought I might point it out. If you set an alarm on your phone for the same time as the Bedtime alarm, alarms may stop working. Just something to be on the lookout for.