Yes. Here's how to set a separate sudo password:

Assume your primary user account is called "user".

Set up a second account, whose password you want to be your sudo password. Let's call this account "admin".

In /etc/sudoers:

  Defaults:user runas_default=admin
  Defaults:user runaspw

Now, as "user", you can

  sudo -u root whoami

and you will be asked for "admin"'s password.

The only downside is that "root" is no longer the default user to become, you have to explicitly specify it.

Hm... would the user not being in sudoers and using su with root password being different than the user password make more sense?