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To show that the limit is complex infinity it is sufficient to show that the modulus goes to infinity

y=var('y',domain='real')
limit(cos(x+I*y).abs(),y=+oo,algorithm='sympy') 
+Infinity

To show that the limit is complex infinity it is sufficient to show that the modulus goes to infinity

y=var('y',domain='real')
limit(cos(x+I*y).abs(),y=+oo,algorithm='sympy') 
+Infinity

Note that |cos(x+I*y)|^2=cos^2(x)+sinh^2(y).