For the tzatziki: Combine the yogurt, mayonnaise, basil, chives, tarragon, vinegar, salt and pepper in a food processor. Blend until the herbs are finely chopped and the sauce is pale green. Scrape the sauce into a bowl.
Coarsely grate the cucumber using the large holes of a box grater onto a paper-towel-lined-plate; pat dry. Mix the cucumber into the tzatziki. Add salt and pepper to taste. The tzatziki can be made 4 hours ahead, covered and chilled, and stirred before serving.
For the salmon: Sprinkle the salmon with salt and pepper. Heat the oil over medium-high heat in a large skillet. Cook the salmon, skinned-side-up, until golden brown, about 4 minutes. Flip the fish using a fish spatula and cook for 2 minutes for medium doneness.
Serve warm, at room temperature or chilled with the tzatziki. To serve chilled, cool, cover, and chill up to 1 day.
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