Question: > 
> It sounds like that she started cooking
> 
> 
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versus 

> 
> It sounds like she started cooking.
> 
> 
> 

1- Are these the same or different?
2- Are they complex sentences?

Answer: 
> 
> [1]\* *It sounds like* [*that* *she started* [*cooking*]].
> 
> 
> [2] *It sounds like* [*she started* [*cooking*]].
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> 
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[1] is ungrammatical because with prepositional governors such as "like", only non-expandable content clauses are permitted, i.e. the subordinator "that" is inadmissible here. 

[2] is a complex sentence for it has an independent clause (the sentence as a whole) and two embedded subordinate clauses consisting of a content clause (in outer brackets) and an embedded gerund-participial clause (in inner brackets).

(Note that if "cooking" is interpreted as a noun, which is possible, then there is only one subordinate clause).
Both "like" and "that" can function as subordinating conjunctions.  Using both to introduce a single subordinate clause is an error.  

"That" doesn't co-locate well with the verbs "sounds" and "looks", although it appears regularly alongside similar verbs such as "appears" and "seems".  

Yes, "It sounds like she started cooking." is a complex sentence.  There is a subordinate clause contained within another clause.  "She started cooking" could stand as a sentence on its own, but "like she started cooking" needs something to modify.  In this case, it acts as a predicative adjectival subject complement, modifying the dummy subject as licensed by the copular verb.