Is this a complex sentence or a compound sentence?

You are correct: this is a complex sentence, not a compound one.

However, the punctuation for the sentence seems a bit iffy, and the correct punctuation would be "Such birds do us good, though we no longer take omens from their flight on this side and that; even the most superstitious villagers no longer take off their hats to the magpie and wish it good-morning."

In particular, the "and" right after the semicolon is unnecessary and informal, not to mention misleading. In fact, the purpose of a semicolon is to remove the necessity of conjunctions to connect independent clauses in the first place, but I suppose the author added the conjunction there for the purpose of effect or clarity.

This appears to be two independent clauses joined by a semicolon—making it a compound sentence. If punctuated as two separate sentences, it would look like this:

Such birds do us good, though we no longer take omens from their flight on this side and that. And even the most superstitious villagers no longer take off their hats to the magpie and wish it good-morning.

The use of the semicolon in this specific case (because of its context) makes it awkward, but not actually incorrect.

From TERMIUM Plus:

There are three ways of joining independent clauses into a compound sentence:

      with a coordinating conjunction (one of the fanboys);
      with a semicolon; or
      with a semicolon and a transitional expression.

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As for starting the second independent clause with and, Neal Whitman says this in "Can I Start a Sentence with a Conjunction?":

Here’s what some of the big usage guides say on the matter. The one that seems to get quoted the most is the Chicago Manual of Style, which says:

There is a widespread belief—one with no historical or grammatical foundation—that it is an error to begin a sentence with a conjunction such as and, but or so. In fact, a substantial percentage (often as many as 10 percent) of the sentences in first-rate writing begin with conjunctions. It has been so for centuries, and even the most conservative grammarians have followed this practice.

Both Garner’s Modern American Usage, and Fowler’s Modern English Usage call this belief a superstition. The Merriam-Webster Dictionary of English Usage (or MWDEU) says, “Everybody agrees that it’s all right to begin a sentence with and,” and notes that you can find examples of it all the way back to Old English.

For better or worse, it seems as if the author decided to make use of an initial and in the second independent clause in addition to using a semicolon rather than a period. The combination of these two things is nonstandard and uncommon.