submitbutton() is deprecated and internally use Joomla.submitbutton .. so you got recursion,
Correct code is:

Joomla.submitbutton = function(task) {
 // some of your code 
 ...
 ...
 Joomla.submitform(task); // actually submit the form
}

Nice... So I should update all my components due to this doubtful "deprecation". Very smart.

Will Joomla.submitform(task) work for older versions of Joomla or should I add another version check just for this?

Sorry for disappoint you, but submitbutton() deprecated since Joomla 2.5 and Joomla.submitbutton , Joomla.submitform added since then .. or maybe even in Joomla 1.6 ..
Also your original code is incorrect ...

so I wonder why you wonder

I wonder, because it worked perfectly upto 3.4.3.

Joomla.submitbutton and Joomla.submitform are indeed present in 1.6.0 ;)

there was some changes in core.js in 3.4.x,
I guess that changes have uncovered your bug

I know it is a bit confused, because there no explanation,
but correct usage is to call submitform() inside submitbutton() .. but do not call submitbutton() inside submitbutton() ...
no matter it is Joomla.submitbutton() (Joomla.submitform()) or old submitbutton() (submitform())

Thank you @Fedik for the explanation. I will close this issue since the code was used incorrectly even before the changes in the core.js. @thekrotek Please update the code in your extensions!