In the case of your example, the authentication is done by ajax and does not execute the form Submit, preventing the browser from storing the password.
For other cases, you can use the autocomplete='off'
in the inputs.
<input type='text' name='username' autocomplete='off'>
Chrome forcing the bar
Alternative 1
To convert the case of Chrome to force the password save, you can do something like this by passing any string instead of off in the autocomplete.
<input type="text" name="field" autocomplete="nope">
Alternative 2
Another alternative is to include an invisible password above the real password, Google Chrome applies this rule only for the first input, so your will be safe from it.
<input style="opacity: 0;position: absolute;">
<input type="password" style="opacity: 0;position: absolute;">
Related: Do not let browser save password
– Guilherme Nascimento