First of all (about categories of objects in German grammar)
There are no direct or indirect objects in German grammar! Forget these terms, they are misleading. And tell whoever taught you these terms, that he/she should stop teaching things that don't exist. Thinking in these categories maybe brings you good results in 90% of all sentences, but in 10% you are wrong! Take any German grammar book and look for direct and indirect objects: You will not find them. If you think in the really existing categories, then you have a good chance to get 100% of all sentences correct.
German has:
- genitive objects
- dative objects
- accusative objects
- prepositional objects
- parts of speech that at first sight look like nominative objects (Gleichsetzungsnominativ)
But there is no direct object and there is no indirect object!
Answer to your question
The verb suchen can be used with an accusative object:
Ich suche ein Zimmer.
I'm looking for a room.
This is the simplest usage of suchen. You don't need any preposition here. The part "ein Zimmer" is an accusative object. Other cases would be wrong:
- genitive case is wrong:
Ich suche eines Zimmers.
- dative case is wrong:
Ich suche einem Zimmer.
Nominative case is wrong too, but for "ein Zimmer" nominative and accusative looks identical. But there are nouns where it makes a difference:
I'm looking for a raven.
- nominative case is wrong:
Ich suche ein Rabe.
- accusative case is correct: Ich suche einen Raben.
The verb suchen allows you to replace the accusative object with a prepositional object:
Ich suche nach einem Zimmer.
I'm looking for a room.
The part "nach einem Zimmer" is a prepositional object. It consists of a preposition (nach) and a nominal group (einem Zimmer). The grammatical case of this internal nominal group (it is inside the prepositional object) is not ruled by the verb (which is outside), but by the preposition (which is the head of the prepositional object). And the preposition "nach" always needs its nominal group to be in dative case. So this is wrong:
- nominative case is wrong:
Ich suche nach ein Zimmer.
- genitive case is wrong:
Ich suche nach eines Zimmers.
- accusative case is wrong:
Ich suche nach ein Zimmer.
After nach only
- dative case is correct: Ich suche nach einem Zimmer.
But you can also add additional prepositional objects:
- without additional prepositional object:
Ich suche den Brief = Ich suche nach dem Brief.
I'm looking for the letter.
- with additional prepositional object:
Ich suche den Brief für meinen Chef. = Ich suche für meinen Chef nach dem Brief.
I'm looking for the letter for my boss.
The part für meinen Chef (engl: for my boss) is an optional prepositional object. It tells you who benefits from the search.
You also can tell where you are looking for something:
Ich suche den Brief in der Küche. = Ich suche in der Küche nach dem Brief.
I'm looking for the letter in the kitchen.
Ich suche den Brief vor dem Haus. = Ich suche vor dem Haus nach dem Brief.
I'm looking for the letter in front of the house.
Ich suche den Brief unter dem Tisch. = Ich suche unter dem Tisch nach dem Brief.
I'm looking for the letter under the table.
This way you can combine suchen with almost any preposition.
In German you also can omit the part of speech that tells you what you are looking for:
Ich suche.
I'm searching.
But still you can add any of the optional prepositional objects:
Ich suche für meinen Chef.
I'm looking for something for my boss.
also:
Ich suche unter dem Tisch. Ich suche vor dem Haus. Ich suche in der Küche.
In "Ich suche für meinen Chef" you do not tell that the thing/person you are looking for is your boss. You tell, that - whatever you are looking for - will result in a benefit for the boss.
You also can search for things that might give your room a better look:
Ich suche einen Teppich für mein Zimmer. = Ich suche nach einem Teppich für mein Zimmer.
I'm looking for a carpet for my room.
So, when you omit that you are after a carpet, you just can leave it out:
Ich suche für mein Zimmer.
I'm looking for something for my room.
This is not very common, but possible and correct. So, when you make a wrong translation, and say:
Ich suche für ein Zimmer mit Bad.
Then you say, that you are looking for something that might improve an apartment with bathroom. But you do not tell what you are looking for. (Note, that in this sentence it makes more sense to translate Zimmer as apartment.)
This is a possible conversation in a home-center:
- German
Verkäufer: "Guten Tag, kann ich Ihnen helfen?"
Kundin: "Guten Tag, ja gerne. Ich interessiere mich für einen dieser Teppiche."
Verkäufer: "Wie groß soll er sein? In welchem Raum soll er liegen?"
Kundin: "Ich suche für ein Zimmer mit Bad."
- English translation
Shop assistant: "Hello, can I help you?"
Customer: "Hello, yes. I'm interested in one of these carpets."
Shop assistant: "How big should it be? In which room should it lie?"
Customer: "I'm looking for something for an apartment with a bathroom."
Compare to this conversation
- German
Wohnungsmakler: "Guten Tag, wie kann ich Ihnen helfen?"
Kundin: "Guten Tag, ich suche nach einem Zimmer mit Bad."
- English
accommodation broker: "Hello, how can I help you?"
Customer: "Hello, I'm looking for an apartment with a bathroom."