Why a clock on sea beach tells a different time than a clock at mountain peak?

There’s relative time dilation between a clock on the beach and mountain due to differences in the strength of the gravitational fields. A clock in higher gravity runs slower than a clock in lower gravity.

That is why we call the real time .

different gravity

(is also different between upstairs and downstairs in a house)

Because stronger gravity slows time.  However, the difference would be absolutely minute.

If they are in different time zones, of course they will be different.

Atomic clocks are accurate anywhere human will ever be.

They work by dividing the trillions of vibrations down to nanoseconds.

Divide that by 1e+9, you get seconds etc...etc..

Wrong,,the notion time dilates with gravitational fields.

That takes a presession and show me a clock that has that.

A magnetometer is based on cesium precession.

Because Heidi's grandfather had a Swiss made atomic clock which is more accurate.

The closer you are to a gravitational source (like the Earth), the slower time will run.  You & I, on Earth's surface, will age a tiny bit slower than the people orbiting us on the space station.   Although the amount of difference is less, time for someone on a mountain peak will run a tiny bit faster than for someone at sea level. 

Time runs slower the deeper in gravity well the clickbis than the clock at a higher altitude because of general relativity and time dilation. This assumes the clicks are at the same longitude and real give my close to each other on longitude. How far Earth is from the Sun also makes a slight difference in the rate that radioactive iOS tropes decay . Because Earth is orbiting around the Sun and is moving,  special relativity is in evolved and can have the opposite effect. 

One clock broken.

Maybe seawater got in, or dropped while scaling the mountain.

Prophet say: No humor allow in astronomy. Apparently.