Question

Is the Sentence

I'm doing dishes and its so boring .

A Incomplete Sentence , or A Complete Sentence ?

Answer 1

incomplete sentence since there are two topics you are talking about

Answer 2

Other than it has a small error, there's nothing the matter with it.
I'm doing dishes, and it's so boring. I guess there are 2 small errors. 

The sentence is complete. There are two main clauses
They can stand alone.
I'm doing dishes.
It's so boring.