Both "chase" and chase after" can be used, but not equally with "around the house." When you use "chase after" you don't usually use "around" + a noun. Google results (where * stands for "any word," in this case the direct object of the verb):
"Chased after * around the" = 30
"Chased * around the" = 5,860
'"Chased after * around the house" yielded one example.
The Merriam-Webster Online Dictionary, athttp://www.m-w.com/cgi-bin/dictionary
...gives this information:CHASE
Inflected Form(s): chased ; chas·ing
1 a : to follow rapidly :PURSUE b:HUNT c:to follow regularly or persistently with the intention of attracting or alluring
3: to seek out -- often used with down <detectives chasing down clues>
4: to cause to depart or flee
RIVE <chase the dog out of the garden>
5: to cause the removal of (a baseball pitcher) by a batting rallyintransitive senses
1: to chase an animal, person, or thing < chase after material possessions
The more natural sentence, then, is Sentence 2.