I do not like you, I like Monika.
Is the sentence above okay? My concern is that there is no conjunction word between the two clauses.
Or should it instead be broken into two separate sentences?
I do not like you. I like Monika.
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The quote is just fine with the comma. It's really up to you, but as Clare mentioned, it's OK or even preferable to use a comma when both phrases are short and meant to be read without much of a pause between the two.
If you use periods instead of a comma, it implies that there is more of a pause between the sentences.
You could use periods for dramatic effect in casual writing:
Best. Party. Ever.
A famous Latin quote can be written as follows:
He came, he saw, he conquered.
You could also use an emdash. This might be more typical in fiction-writing or casual writing. The emdash is very flexible and useful in situations like this:
I don't like you -- I like Monika.
A more formal or old-fashioned style would be to use a semicolon:
I don't like you; I like Monika.