it's a verb

ich schaue mich is a proper sentenc

with a subject

ich

a verb

schaue

and an object

so the object has to follow grammatical rules

ich schaue mich is not a proper sentence
ich schaue mich an would be

but thats not what he is asking

its about viewing an object

Ich schaue mir das Haus an
Er schaut sich das Haus an

and no I have no idea why that is

what

ich schaue mich

i see myself

I look at myself rather

ich sehe mich

anschauen

ich schaue mich an

yeah ok i forgot the preposition

ich schaue mich an

ich schaue mir is a completely different thing

ja

well ok

with akkusativ it's just imperativ

mir/mich
dir/dich
?/sich

i dont think there is anything there

"Ich schaue mir X" means "I look at X for my own pleasure"

it's the same thing we talked about

like

yesterday

look, for your pleasure, at my city

but whats dir doing there

schau meine stadt an would technically be correct as well

but including dir gives it a sound of strolling through it experiencing it first hadn

like dave said, for your own pelasure

its a decent thought to keep in mind

it would also work on the opposite sense, mind you

for example