Bdtyktl

Some answers are outdated in this thread.
To install multiple clocks, you need gnome-clocks which is now available in the official repos.

(Note: The button above points to the snap store, which does have gnome-clocks. But the snap application doesn't fully work on Ubuntu 18.04 LTS, the times don't show up.)

Or, to do it from the command line:

1. Open a terminal(CtrlAltT)


2. Fire sudo apt install gnome-clocks to grab them!

Logout and login after installing and it will appear in your notification menu.

I found gnome-clocks on Launchpad. It can be installed with:

sudo apt-get update

sudo apt-get install gnome-clocks 

As of 18.04 LTS. It looks like this:

Today there is MultiClock, a Gnome 3 Extension that puts another clock into your bar.

Really useful and allow multiple timezones.

• https://extensions.gnome.org/extension/605/multiclock/


• Source: https://github.com/mibus/MultiClock
  

Only drawback is you need to know how to edit JavaScript for its configuration:

gedit ~/.local/share/gnome-shell/extensions/MultiClock@mibus.org/extension.js

But it's not hard. In the file change the following variables according to your timezone parameters

const Timezones = {

    'UTC': { hr: 0, min: 0, tzname: 'UTC' },

    'Australia/Adelaide': { hr: 10, min: 30, tzname: 'ACDT' },

    'Australia/Perth': { hr: 8, min: 0, tzname: 'WST' },

};

This GNOME extension called Top Bar Clocks lets the user add multiple clocks to the top bar.

Have you tried the clock applet from the Screenlets? You can have multiple clocks set to different time zones.
You can install screenlet by doing sudo apt-get install screenlets in terminal.

You can add/remove individual clocks. And set individual/all clocks to autostart at login.

I use FoxClocks, which is an add-on for Firefox.

While it is not in Unity/Gnome, I normally have Firefox open anyhow.

I agree with the solutions posted above (specifically gnome-clocks). I just wanted to add one thing (and I do not have enough reputation to comment). You actually do not need to log out and log back in, it should be sufficient if you just restart gnome (ALT + F2, type r and [ENTER], your shell should restart and additional clocks should appear)

An extension that puts different clocks directly on the top bar is Panel Work Clock Lite. It appears to be well maintained as of the writing of this answer.

Timezone is a beautiful GNOME extension to see people with their timezones. More info see at the github-page.

1. Install the extension (when your GNOME Shell is ready for it)

2. Create a text-file with the name people.json (*example code see below)

3. Place the file where you like.

4. Edit the settings of the extensions in Ubuntu Tweak or directly on the extension-page when you have the Firefox Gnome extension add-on and set where the json-file is placed, e.g. file:///home/USERNAME/people.json.

5. Logout & login or Alt-F2 with command r (or maybe a restart is needed)

*Example code for people.json:

[

          {

            "name": "California",

            "avatar": "file:///home/USERNAME/Pictures/California.JPG",

            "city": "Los Angeles",

            "tz": "America/Los_Angeles"

          },

          {

            "name": "Hans Pacquiao",

            "avatar": "file:///home/cds/Pictures/Hans.JPG",

            "city": "Indiana",

            "tz": "America/Indiana/Knox"

          },

          {

           "name": "Samar friends",

           "avatar": "file:///home/USERNAME/Pictures/Samarfriends.ico",

           "city": "Philippines",

           "tz": "Asia/Manila"

          }

]

For more time zones information, see wikipedia.