Direct answer

You cannot really pass the bash array as a list inside an extra var to ansible. It is possible but would require to loop over the bash array and transform it into a json parsable format that you inject inside the extra var.

The easiest way IMO is to pass a concatenation of all array element inside a string that you will later split in the playbook.

Using the form ${somevar[@]} won't work here as every bash array element will end up being parsed as a new argument to your ansible command. You will have to use instead ${somevar[*]}. You also need to quote the extra var correctly so that both bash and ansible are able to successfully parse it. The correct command call in your script is then:

ansible-playbook test.yml --extra-vars "ntp_list_raw='${ntpServers[*]}'"

You now need a bit of rework on your ansible playbook to split the received value into a list:

---
- name: "This is a test"
  hosts: localhost
  gather_facts: no

  vars:
    ntp_list: "{{ ntp_list_raw.split(' ') }}"

  tasks:

    - name: Print split variable items
      debug:
        var: item
      loop: "{{ ntp_list }}"

And now entering values and calling the playbook from your script gives:

Please input the ip address or domain name of the TP server you wish to add: toto
Please input the ip address or domain name of the TP server you wish to add: pipo
Please input the ip address or domain name of the TP server you wish to add: bingo
Please input the ip address or domain name of the TP server you wish to add: 

PLAY [This is a test] *********************************************************************

TASK [Print split variable items] *********************************************************************
ok: [localhost] => (item=toto) => {
    "ansible_loop_var": "item",
    "item": "toto"
}
ok: [localhost] => (item=pipo) => {
    "ansible_loop_var": "item",
    "item": "pipo"
}
ok: [localhost] => (item=bingo) => {
    "ansible_loop_var": "item",
    "item": "bingo"
}

PLAY RECAP *********************************************************************
localhost                  : ok=1    changed=0    unreachable=0    failed=0    skipped=0    rescued=0    ignored=0 

Going further

If your only goal is to ask for your ntp servers interactively, you can do all of the above directly in your playbook using vars_prompt

---
- name: "This is a test"
  hosts: localhost
  gather_facts: no

  vars_prompt:
    - name: ntp_list_raw
      prompt: "Please enter ntp servers separated by a comma without spaces"
      private: no

  vars:
    ntp_list: "{{ ntp_list_raw.split(',') }}"

  tasks:

    - name: Print split variable
      debug:
        var: item
      loop: "{{ ntp_list }}"

which gives:

$ ansible-playbook test.yaml 
Please enter ntp servers separated by a comma without spaces []: toto,pipo,bingo

PLAY [This is a test] *********************************************************************

TASK [Print split variable] *********************************************************************
ok: [localhost] => (item=toto) => {
    "ansible_loop_var": "item",
    "item": "toto"
}
ok: [localhost] => (item=pipo) => {
    "ansible_loop_var": "item",
    "item": "pipo"
}
ok: [localhost] => (item=bingo) => {
    "ansible_loop_var": "item",
    "item": "bingo"
}

PLAY RECAP *********************************************************************
localhost                  : ok=1    changed=0    unreachable=0    failed=0    skipped=0    rescued=0    ignored=0

You can even bypass the prompt giving the value as an extra var directly:

$ ansible-playbook test.yaml -e ntp_list_raw=toto,pipo,bingo

PLAY [This is a test] *********************************************************************

TASK [Print split variable] *********************************************************************
ok: [localhost] => (item=toto) => {
    "ansible_loop_var": "item",
    "item": "toto"
}
ok: [localhost] => (item=pipo) => {
    "ansible_loop_var": "item",
    "item": "pipo"
}
ok: [localhost] => (item=bingo) => {
    "ansible_loop_var": "item",
    "item": "bingo"
}

PLAY RECAP *********************************************************************
localhost                  : ok=1    changed=0    unreachable=0    failed=0    skipped=0    rescued=0    ignored=0