You need to look for a different application. I use DayOne, and it shows the time of the entry, plus the timezone of the post. So, for example, a post I made while in London indicates it was made at 11:41 AM, GMT+1. Plus, all my images (in Photos) taken with my iPhone show the local time of taking the picture, not the time of where I am a the current time. The only "shifting" time is in Calendar, as it should be.
When I got home, the date (and time) for the entry had changed to the date and time it was back here on the east coast (a day later!). I contacted the developer and asked for a switch that would allow the time/date for an entry to stay what it was locally, but they -- and, it would seem, all apps that use date/time tagging -- have decided that we always want the time to float to whatever time zone we travel to in the future.