
ausschauen
to look
To have a certain appearance or give a certain impression. Used especially in southern Germany and Austria.
Mit dem neuen Haarschnitt schaust du wirklich gut aus.
You look really good with the new haircut.
to look / to look out
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to look
To have a certain appearance or give a certain impression. Used especially in southern Germany and Austria.
Mit dem neuen Haarschnitt schaust du wirklich gut aus.
You look really good with the new haircut.

to look out for
To search specifically for someone or something, or to keep your eyes open in order to find it.
Wir sollten nach einem günstigen Hotel für unseren Urlaub Ausschau halten.
We should look out for an affordable hotel for our vacation.
'Ausschauen' is a separable verb with the prefix 'aus-'. In a main clause, 'aus' goes at the end of the sentence.
Du schaust heute aber müde aus.
You look pretty tired today.
In the meaning 'aussehen' (to look), 'ausschauen' is used especially in southern Germany, Austria, and Switzerland. In other regions of Germany, you hear it less often.
Wie schaut's aus? Schaffen wir den Zug noch?
How does it look? Will we still make the train?
When 'ausschauen' means 'aussehen' (to look), it is almost always combined with an adjective or adverb that describes the condition or appearance.
Das Kleid schaut an dir fantastisch aus.
That dress looks fantastic on you.
The phrase 'Ausschau halten nach' (to look out for) is a fixed combination. Here, 'Ausschau' is a noun. The preposition 'nach' always takes the dative.
Halte bitte nach einem freien Parkplatz Ausschau.
Please look out for a free parking space.
to search / look for
Suchen generally refers to the active attempt to find someone or something. Here, ausschauen means the expression Ausschau halten nach: you watch your surroundings attentively and wait to spot the person or thing you are looking for.

Ich halte am Eingang des Bahnsteigs nach Anna Ausschau.
I am keeping an eye out for Anna at the entrance to the platform.
Ich suche auf dem ganzen Bahnsteig nach Anna.
I am looking for Anna all along the platform.