ich bin einkaufen
Is the sentence correct? -- I am shopping
Found this link that confirms it. This sentence is ok when not present. So although it's not the most common sentence, it's not incorrect. See more on the German Absentiv if you don't believe it :)
No, "Ich bin einkaufen." is actually one of the most natural ways to say this. "am Einkaufen" or "beim Einkaufen" relates to the act of shopping but "Ich bin einkaufen." is a short form for: "Ich bin weg/aus Haus, um einzukaufen." = "I am away/out of the house to shop." You relate "einkaufen" to the place and the purpose. Another verb this is done with is "gehen". "Ich gehe schwimmen.", "Ich gehe tanzen.", "Ich gehe feiern." meaning: "I am leaving for matters of swimming/dancing/celebrating." or more common: "I go swimming/dancing/celebrating." So while "gehen" looks at the activities as something were you have to walk to, "sein" acts here to say that you already arrived and are now in the act of it.
It's a bit more restricted than that -- read the Wikipedia article.
This form more or less expresses that you are not currently in the expected location because you are away doing something else but will be back.
"Ich bin denken" sounds a bit odd because people usually don't go to a specific place in order to think.
"Sie ist joggen" or "Er ist einkaufen" or so would work better.