My 2 month old baby cries a lot. Could this be some kind of allergy?

Excessive crying is commonly seen in CMPA

YES. This is a possibility

It could be COW'S MILK PROTEIN ALLERGY (CMPA)

Things to look out for that could indicate CMPA:

1) excessive crying episodes especially when drinking milk/feeding. Arching their back frequently with these crying episodes.

2) frequent vomiting episodes.

3) frequent loose stools.

4) Blood or mucous in stools.

5) Frequent eczema flare ups or eczema that is difficult to treat.

6) Eventually, it may lead to your child faltering in growth.

This kind of allergy is generally Non - IgE mediated allergic reaction. Please see the introduction page for more information about Non-Ige mediated allergic reactions.

CAUTION - if your child comes up with a red blotchy, pin prick like rash after drinking cow's milk/cow's milk products - this could be a sign of IgE mediated allergy. Please seek an urgent appointment with a Paediatric Allergist.
Please see the introduction page for more information about Ige mediated allergic reactions.

If your child has the  above mentioned signs with raised temperature/fever, then the first thing to rule out would be an underlying infection for which please seek an appointment with your Paediatrician urgently.

What has allergy got to do with these crying episodes?

Cow's milk protein allergy leads to inflammation in your baby's gut. 
This inflammation leads to pain and crying episodes. 
A hallmark of this problem is a start of crying episode during feeding with frequent arching of baby's back.

How common is cow's milk protein allergy?

CMPA affects about 7% of formula-fed babies but only about 0.5% of exclusively breast-fed babies, who also tend to have milder reactions.
Exclusive breast-feeding may also protect babies from developing an allergy to cow's milk protein after they are weaned.

Reference - Liao SL, Lai SH, Yeh KW, et al; Exclusive breastfeeding is associated with reduced cow's milk sensitization in early childhood. Pediatr Allergy Immunol. 2014 Aug25(5):456-61. doi: 10.1111/pai.12247.

Could this be lactose intolerance?

The symptoms and signs highlighted above are HIGHLY UNLIKELY to be related to lactose intolerance.It is generally a condition of older childhood.

Lactose intolerance is NOT AN ALLERGY .
Lactose intolerance is the inability of a person's gut to digest lactose (sugar) in the milk because of an inherited deficiency of an enzyme called as lactate is in the individual's gut.  Symptoms are caused by the undigested lactose in the gut.

The common symptoms of lactose intolerance are:
1) your child may get bloating or stomach pains after drinking milk or milk products.
2) passing a lot of wind or may have watery diarrhea.

How can I manage my child's cow's milk protein allergy ?

The primary objective would be to avoid the protein in cow's milk that is triggering these allergic reactions.

If you have been breast-feeding your child and would like to continue that then, you must eliminate cow's milk and cow's milk products from your diet. This also includes avoiding anything that is derived from cow's milk such as casein, whey and lactose. 

Breast-feeding mums who avoid cows milk products , should ensure that they have enough calcium in the diet. 
It is advisable that you should take calcium and vitamin D tablets ( a daily recommendation of around 1000mg calcium and 10 mcg Vitamin D).
You can also increase your calcium intake in your diet by including tinned fish, pulses, almonds and oranges. 
Iodine is often lacking in the diet of people on Milk/Dairy free diets. Rich sources of Iodine are fish and seaweed.

If your child is formula fed, they can be given extensively hydrolyzed/amino acid formula milk (specialised milk formulas). This formula milk comes from cow's milk but the cow's milk protein is broken down into small peptides and amino acids that are less likely to trigger an allergic reaction.
Please discuss with your Paediatric allergist, what is the optimal extensively hydrolyzed/amino acid formula milk formula for your child.

There is some evidence that hydrolyzed formula milks can provide a balanced nutrition and restore normal growth and development in children who have their growth and development impaired by cow's milk protein allergy*.

Babies who are being weaned and older children with persisting cow's milk protein allergy will need to follow a cow's milk free diet.

Reference -
* Dupont C, Hol J, Nieuwenhuis EE; An extensively hydrolysed casein-based formula for infants with cows' milk protein allergy: tolerance/hypo-allergenicity and growth catch-up. Br J Nutr. 2015 Apr 14113(7):1102-12. doi: 10.1017/S000711451500015X.

Can I give my child any other milk rather than the specialized formula milk?

Soya milk can be introduced into your child's diet when they are around a year old. Soya milk contains isoflavones which have a weak oestrogenic (a hormone that is a primary female sex hormone) activity and hence it's not advised till children are around one year-old.
Caution - About 15% of children who are allergic to cow's milk will also react to soya.

Coconut milk, Oat milk and other Nut based milks may be used after the age of 2 years.

Rice milk can be given to children who have cow's milk allergy aged 5 years and above.

Goat's milk, Sheep milk or other mammal milks are not suitable substitutes as the protein in their milk is quite similar to the cow's milk protein.

Will my child outgrow cow's milk protein allergy?

IMAP milk ladder for cows milk challenge in children with non IgE mediated CMPA

The outlook with cow's milk protein allergy is generally favourable with
approximately 45-50% outgrowing it at 1 year, 60-75% at 2 years and 85-90% at 3 years of age(1).

Children can have a challenge test at home starting from when they are around 1 year old (provided they have Non IgE mediated cow's milk protein allergy). Children can have a challenge test roughly every 6 months to see whether they can tolerate milk using a 'MILK LADDER' from iMAP guideline (2)
This is a hierarchy of milk-containing foods, beginning with those least likely to cause a reaction and gradually moving towards being able to drink a glass of milk. Baked form of cow's milk is less allergenic and may be tolerated sooner than unbaked milk.

Please don't consider a challenge to cow's milk at home if your child has an IgE mediated cow's milk protein allergy.

 Reference:

(1) Host A, Halken S; Cow's milk allergy: where have we come from and where are we going? Endocr Metab Immune Disord Drug Targets. 2014 Mar14(1):2-8.

(2)  Venter C, Brown T, Meyer R, et al; Better recognition, diagnosis and management of non-IgE-mediated cow's milk allergy in infancy: iMAP-an international interpretation of the MAP (Milk Allergy in Primary Care) guideline. Clin Transl Allergy. 2017 Aug 237:26. doi: 10.1186/s13601-017-0162-y.

can CMPA be life threatening?

YES! unfortunately, there is well researched evidence to suggest that this is distinctly possible.  

IgE mediated reactions to cows milk protein (sudden onset reactions) can turn out to be  severe allergic reactions which may become life threatening. It is one of the leading causes of death due to allergic reactions in young children.

Reference: Turner PJ, Gowland MH, Sharma V, et al. Increase in anaphylaxis-related hospitalizations but no increase in fatalities: An analysis of United Kingdom national anaphylaxis data, 1992-2012. The Journal of Allergy and Clinical Immunology. 2015;135(4):956-963.e1. doi:10.1016/j.jaci.2014.10.021.