It might be possible with a core hack. In qa-include/app/page.php and qa-include/pages/admin/admin-widgets.php look for the $regioncodes variables and add 'L' => 'leftside' to the array. (Those variables ought to be defined in one place instead of duplicated code but anyway...)
Then, in your advanced theme override the sidepanel() function and add your left sidepanel. Example:
public function sidepanel()
{
$this->output('<div class="qa-sidepanel-left">');
$this->widgets('leftside', 'top');
$this->widgets('leftside', 'bottom');
$this->output('</div>', '');
parent::sidepanel();
}
You could also add high/low if needed. Then in your plugin use 'leftside' as the region name.
A slightly simpler method, without a core hack, would be to use the top/bottom 'place codes' on one side and high/low on the other. You can do a similar theme override to above:
public function sidepanel()
{
$this->output('<div class="qa-sidepanel-left">');
$this->widgets('side', 'top');
$this->widgets('side', 'bottom');
$this->output('</div>', '');
$this->output('<div class="qa-sidepanel">');
$this->widgets('side', 'high');
$this->sidebar();
$this->widgets('side', 'low');
$this->output_raw(@$this->content['sidepanel']);
$this->feed();
$this->output('</div>', '');
}
Then if you want a widget on the left, in the admin you can set it to "Side panel - Top" or "Side panel - Last". If you want it on the right, use "Side panel - Below sidebar box" or "Side panel - Below categories".
This does remove a little bit of flexibility as you can only put widgets in 2 locations, not 4. But you can experiment with moving around the various lines there.